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Theorem idomrootle 27479
Description: No element of an integral domain can have more than  N  N-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
Hypotheses
Ref Expression
idomrootle.b  |-  B  =  ( Base `  R
)
idomrootle.e  |-  .^  =  (.g
`  (mulGrp `  R )
)
Assertion
Ref Expression
idomrootle  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 { y  e.  B  |  ( N 
.^  y )  =  X } )  <_  N )
Distinct variable groups:    y, B    y, N    y, R    y, X
Allowed substitution hint:    .^ ( y)

Proof of Theorem idomrootle
StepHypRef Expression
1 eqid 2435 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2435 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2435 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
4 eqid 2435 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
5 eqid 2435 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
6 eqid 2435 . . 3  |-  ( 0g
`  (Poly1 `  R ) )  =  ( 0g `  (Poly1 `  R ) )
7 simp1 957 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e. IDomn )
8 isidom 16356 . . . . . . . . 9  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
98simplbi 447 . . . . . . . 8  |-  ( R  e. IDomn  ->  R  e.  CRing )
107, 9syl 16 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  CRing )
11 crngrng 15666 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
1210, 11syl 16 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Ring )
131ply1rng 16634 . . . . . 6  |-  ( R  e.  Ring  ->  (Poly1 `  R
)  e.  Ring )
1412, 13syl 16 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (Poly1 `  R )  e.  Ring )
15 rnggrp 15661 . . . . 5  |-  ( (Poly1 `  R )  e.  Ring  -> 
(Poly1 `
 R )  e. 
Grp )
1614, 15syl 16 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (Poly1 `  R )  e.  Grp )
17 eqid 2435 . . . . . . 7  |-  (mulGrp `  (Poly1 `  R ) )  =  (mulGrp `  (Poly1 `  R
) )
1817rngmgp 15662 . . . . . 6  |-  ( (Poly1 `  R )  e.  Ring  -> 
(mulGrp `  (Poly1 `  R
) )  e.  Mnd )
1914, 18syl 16 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (mulGrp `  (Poly1 `  R ) )  e.  Mnd )
20 simp3 959 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  NN )
21 eqid 2435 . . . . . . 7  |-  (var1 `  R
)  =  (var1 `  R
)
2221, 1, 2vr1cl 16603 . . . . . 6  |-  ( R  e.  Ring  ->  (var1 `  R
)  e.  ( Base `  (Poly1 `  R ) ) )
2312, 22syl 16 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (var1 `  R )  e.  (
Base `  (Poly1 `  R
) ) )
2417, 2mgpbas 15646 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (mulGrp `  (Poly1 `  R ) ) )
25 eqid 2435 . . . . . 6  |-  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
2624, 25mulgnncl 14897 . . . . 5  |-  ( ( (mulGrp `  (Poly1 `  R
) )  e.  Mnd  /\  N  e.  NN  /\  (var1 `  R )  e.  (
Base `  (Poly1 `  R
) ) )  -> 
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) ) )
2719, 20, 23, 26syl3anc 1184 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) ) )
28 eqid 2435 . . . . . . 7  |-  (algSc `  (Poly1 `  R ) )  =  (algSc `  (Poly1 `  R
) )
29 idomrootle.b . . . . . . 7  |-  B  =  ( Base `  R
)
301, 28, 29, 2ply1sclf 16669 . . . . . 6  |-  ( R  e.  Ring  ->  (algSc `  (Poly1 `  R ) ) : B --> ( Base `  (Poly1 `  R ) ) )
3112, 30syl 16 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (algSc `  (Poly1 `  R ) ) : B --> ( Base `  (Poly1 `  R ) ) )
32 simp2 958 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  X  e.  B )
3331, 32ffvelrnd 5863 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )
34 eqid 2435 . . . . 5  |-  ( -g `  (Poly1 `  R ) )  =  ( -g `  (Poly1 `  R ) )
352, 34grpsubcl 14861 . . . 4  |-  ( ( (Poly1 `  R )  e. 
Grp  /\  ( N
(.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) )  /\  ( (algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  (
Base `  (Poly1 `  R
) ) )
3616, 27, 33, 35syl3anc 1184 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  (
Base `  (Poly1 `  R
) ) )
373, 1, 2deg1xrcl 19997 . . . . . . . . . 10  |-  ( ( (algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) )  ->  ( ( deg1  `  R
) `  ( (algSc `  (Poly1 `  R ) ) `
 X ) )  e.  RR* )
3833, 37syl 16 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  RR* )
39 0xr 9123 . . . . . . . . . 10  |-  0  e.  RR*
4039a1i 11 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  e.  RR* )
41 nnre 9999 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  RR )
4241rexrd 9126 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR* )
43423ad2ant3 980 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  RR* )
443, 1, 29, 28deg1sclle 20027 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <_  0
)
4512, 32, 44syl2anc 643 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <_  0
)
46 nngt0 10021 . . . . . . . . . 10  |-  ( N  e.  NN  ->  0  <  N )
47463ad2ant3 980 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  <  N )
4838, 40, 43, 45, 47xrlelttrd 10742 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <  N
)
498simprbi 451 . . . . . . . . . . 11  |-  ( R  e. IDomn  ->  R  e. Domn )
50 domnnzr 16347 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
5149, 50syl 16 . . . . . . . . . 10  |-  ( R  e. IDomn  ->  R  e. NzRing )
527, 51syl 16 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e. NzRing )
53 nnnn0 10220 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  NN0 )
54533ad2ant3 980 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  NN0 )
553, 1, 21, 17, 25deg1pw 20035 . . . . . . . . 9  |-  ( ( R  e. NzRing  /\  N  e. 
NN0 )  ->  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) )  =  N )
5652, 54, 55syl2anc 643 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) )  =  N )
5748, 56breqtrrd 4230 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ) )
581, 3, 12, 2, 34, 27, 33, 57deg1sub 20023 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  =  ( ( deg1  `  R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ) )
5958, 56eqtrd 2467 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  =  N )
6059, 54eqeltrd 2509 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  e.  NN0 )
613, 1, 6, 2deg1nn0clb 20005 . . . . 5  |-  ( ( R  e.  Ring  /\  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  (
Base `  (Poly1 `  R
) ) )  -> 
( ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  =/=  ( 0g `  (Poly1 `  R ) )  <-> 
( ( deg1  `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  e.  NN0 )
)
6212, 36, 61syl2anc 643 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) )  =/=  ( 0g `  (Poly1 `  R ) )  <->  ( ( deg1  `  R ) `  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) )  e. 
NN0 ) )
6360, 62mpbird 224 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  =/=  ( 0g `  (Poly1 `  R ) ) )
641, 2, 3, 4, 5, 6, 7, 36, 63fta1g 20082 . 2  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 ( `' ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) " { ( 0g `  R ) } ) )  <_ 
( ( deg1  `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) )
65 eqid 2435 . . . . . . 7  |-  ( R  ^s  B )  =  ( R  ^s  B )
66 eqid 2435 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
67 fvex 5734 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
6829, 67eqeltri 2505 . . . . . . . 8  |-  B  e. 
_V
6968a1i 11 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  B  e.  _V )
704, 1, 65, 29evl1rhm 19941 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
7110, 70syl 16 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
722, 66rhmf 15819 . . . . . . . . 9  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
7371, 72syl 16 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) ) )
7473, 36ffvelrnd 5863 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(eval1 `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  e.  ( Base `  ( R  ^s  B ) ) )
7565, 29, 66, 7, 69, 74pwselbas 13703 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(eval1 `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) : B --> B )
76 ffn 5583 . . . . . 6  |-  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) : B --> B  -> 
( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  Fn  B )
7775, 76syl 16 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(eval1 `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  Fn  B )
78 fniniseg2 5845 . . . . 5  |-  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  Fn  B  -> 
( `' ( (eval1 `  R ) `  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) " { ( 0g `  R ) } )  =  { y  e.  B  |  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( 0g `  R ) } )
7977, 78syl 16 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( `' ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) " { ( 0g `  R ) } )  =  { y  e.  B  |  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( 0g `  R ) } )
8010adantr 452 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  R  e.  CRing )
81 simpr 448 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  y  e.  B )
824, 21, 29, 1, 2, 80, 81evl1vard 19945 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
(var1 `  R )  e.  ( Base `  (Poly1 `  R ) )  /\  ( ( (eval1 `  R
) `  (var1 `  R
) ) `  y
)  =  y ) )
83 idomrootle.e . . . . . . . . . 10  |-  .^  =  (.g
`  (mulGrp `  R )
)
84 simpl3 962 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  N  e.  NN )
8584, 53syl 16 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  N  e.  NN0 )
864, 1, 29, 2, 80, 81, 82, 25, 83, 85evl1expd 19950 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) )  /\  ( ( (eval1 `  R
) `  ( N
(.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ) `  y )  =  ( N  .^  y ) ) )
87 simpl2 961 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  X  e.  B )
884, 1, 29, 28, 2, 80, 87, 81evl1scad 19943 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( (algSc `  (Poly1 `  R ) ) `  X )  e.  (
Base `  (Poly1 `  R
) )  /\  (
( (eval1 `  R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) ) `  y
)  =  X ) )
89 eqid 2435 . . . . . . . . 9  |-  ( -g `  R )  =  (
-g `  R )
904, 1, 29, 2, 80, 81, 86, 88, 34, 89evl1subd 19947 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) )  e.  ( Base `  (Poly1 `  R ) )  /\  ( ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) `  y )  =  ( ( N  .^  y
) ( -g `  R
) X ) ) )
9190simprd 450 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( ( N 
.^  y ) (
-g `  R ) X ) )
9291eqeq1d 2443 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) `  y )  =  ( 0g `  R )  <-> 
( ( N  .^  y ) ( -g `  R ) X )  =  ( 0g `  R ) ) )
93 rnggrp 15661 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9412, 93syl 16 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Grp )
9594adantr 452 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  R  e.  Grp )
96 eqid 2435 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
9796rngmgp 15662 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
9812, 97syl 16 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (mulGrp `  R )  e.  Mnd )
9998adantr 452 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (mulGrp `  R )  e.  Mnd )
10096, 29mgpbas 15646 . . . . . . . . 9  |-  B  =  ( Base `  (mulGrp `  R ) )
101100, 83mulgnncl 14897 . . . . . . . 8  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  N  e.  NN  /\  y  e.  B )  ->  ( N  .^  y )  e.  B )
10299, 84, 81, 101syl3anc 1184 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  ( N  .^  y )  e.  B )
10329, 5, 89grpsubeq0 14867 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( N  .^  y )  e.  B  /\  X  e.  B )  ->  (
( ( N  .^  y ) ( -g `  R ) X )  =  ( 0g `  R )  <->  ( N  .^  y )  =  X ) )
10495, 102, 87, 103syl3anc 1184 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( N  .^  y ) ( -g `  R ) X )  =  ( 0g `  R )  <->  ( N  .^  y )  =  X ) )
10592, 104bitrd 245 . . . . 5  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) `  y )  =  ( 0g `  R )  <-> 
( N  .^  y
)  =  X ) )
106105rabbidva 2939 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  { y  e.  B  |  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( 0g `  R ) }  =  { y  e.  B  |  ( N  .^  y )  =  X } )
10779, 106eqtrd 2467 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( `' ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) " { ( 0g `  R ) } )  =  { y  e.  B  |  ( N 
.^  y )  =  X } )
108107fveq2d 5724 . 2  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 ( `' ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) " { ( 0g `  R ) } ) )  =  ( # `  {
y  e.  B  | 
( N  .^  y
)  =  X }
) )
10964, 108, 593brtr3d 4233 1  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 { y  e.  B  |  ( N 
.^  y )  =  X } )  <_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948   {csn 3806   class class class wbr 4204   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   0cc0 8982   RR*cxr 9111    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   #chash 11610   Basecbs 13461    ^s cpws 13662   0gc0g 13715   Mndcmnd 14676   Grpcgrp 14677   -gcsg 14680  .gcmg 14681  mulGrpcmgp 15640   Ringcrg 15652   CRingccrg 15653   RingHom crh 15809  NzRingcnzr 16320  Domncdomn 16332  IDomncidom 16333  algSccascl 16363  var1cv1 16562  Poly1cpl1 16563  eval1ce1 16565   deg1 cdg1 19969
This theorem is referenced by:  idomodle  27480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-prds 13663  df-pws 13665  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-rnghom 15811  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lsp 16040  df-nzr 16321  df-rlreg 16335  df-domn 16336  df-idom 16337  df-assa 16364  df-asp 16365  df-ascl 16366  df-psr 16409  df-mvr 16410  df-mpl 16411  df-evls 16412  df-evl 16413  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-evl1 16572  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971  df-mon1 20045  df-uc1p 20046  df-q1p 20047  df-r1p 20048
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