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Theorem idomrootle 27181
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 2388 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2388 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2388 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
4 eqid 2388 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
5 eqid 2388 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
6 eqid 2388 . . 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 16292 . . . . . . . . 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 15602 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
1210, 11syl 16 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Ring )
131ply1rng 16570 . . . . . 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 15597 . . . . 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 2388 . . . . . . 7  |-  (mulGrp `  (Poly1 `  R ) )  =  (mulGrp `  (Poly1 `  R
) )
1817rngmgp 15598 . . . . . 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 2388 . . . . . . 7  |-  (var1 `  R
)  =  (var1 `  R
)
2221, 1, 2vr1cl 16539 . . . . . 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 15582 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (mulGrp `  (Poly1 `  R ) ) )
25 eqid 2388 . . . . . 6  |-  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
2624, 25mulgnncl 14833 . . . . 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 2388 . . . . . . 7  |-  (algSc `  (Poly1 `  R ) )  =  (algSc `  (Poly1 `  R
) )
29 idomrootle.b . . . . . . 7  |-  B  =  ( Base `  R
)
301, 28, 29, 2ply1sclf 16605 . . . . . 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 5811 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )
34 eqid 2388 . . . . 5  |-  ( -g `  (Poly1 `  R ) )  =  ( -g `  (Poly1 `  R ) )
352, 34grpsubcl 14797 . . . 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 19873 . . . . . . . . . 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 9065 . . . . . . . . . 10  |-  0  e.  RR*
4039a1i 11 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  e.  RR* )
41 nnre 9940 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  RR )
4241rexrd 9068 . . . . . . . . . 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 19903 . . . . . . . . . 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 9962 . . . . . . . . . 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 10683 . . . . . . . 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 16283 . . . . . . . . . . 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 10161 . . . . . . . . . 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 19911 . . . . . . . . 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 4180 . . . . . . 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 19899 . . . . . 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 2420 . . . . 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 2462 . . . 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 19881 . . . . 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 19958 . 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 2388 . . . . . . 7  |-  ( R  ^s  B )  =  ( R  ^s  B )
66 eqid 2388 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
67 fvex 5683 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
6829, 67eqeltri 2458 . . . . . . . 8  |-  B  e. 
_V
6968a1i 11 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  B  e.  _V )
704, 1, 65, 29evl1rhm 19817 . . . . . . . . . 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 15755 . . . . . . . . 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 5811 . . . . . . 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 13639 . . . . . 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 5532 . . . . . 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 5793 . . . . 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 19821 . . . . . . . . . 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 19826 . . . . . . . . 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 19819 . . . . . . . . 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 2388 . . . . . . . . 9  |-  ( -g `  R )  =  (
-g `  R )
904, 1, 29, 2, 80, 81, 86, 88, 34, 89evl1subd 19823 . . . . . . . 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 2396 . . . . . 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 15597 . . . . . . . . 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 2388 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
9796rngmgp 15598 . . . . . . . . . 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 15582 . . . . . . . . 9  |-  B  =  ( Base `  (mulGrp `  R ) )
101100, 83mulgnncl 14833 . . . . . . . 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 14803 . . . . . . 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 2891 . . . 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 2420 . . 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 5673 . 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 4183 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 1649    e. wcel 1717    =/= wne 2551   {crab 2654   _Vcvv 2900   {csn 3758   class class class wbr 4154   `'ccnv 4818   "cima 4822    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   0cc0 8924   RR*cxr 9053    < clt 9054    <_ cle 9055   NNcn 9933   NN0cn0 10154   #chash 11546   Basecbs 13397    ^s cpws 13598   0gc0g 13651   Mndcmnd 14612   Grpcgrp 14613   -gcsg 14616  .gcmg 14617  mulGrpcmgp 15576   Ringcrg 15588   CRingccrg 15589   RingHom crh 15745  NzRingcnzr 16256  Domncdomn 16268  IDomncidom 16269  algSccascl 16299  var1cv1 16498  Poly1cpl1 16499  eval1ce1 16501   deg1 cdg1 19845
This theorem is referenced by:  idomodle  27182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-ofr 6246  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-fz 10977  df-fzo 11067  df-seq 11252  df-hash 11547  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-prds 13599  df-pws 13601  df-0g 13655  df-gsum 13656  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-mhm 14666  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mulg 14743  df-subg 14869  df-ghm 14932  df-cntz 15044  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-cring 15592  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-rnghom 15747  df-subrg 15794  df-lmod 15880  df-lss 15937  df-lsp 15976  df-nzr 16257  df-rlreg 16271  df-domn 16272  df-idom 16273  df-assa 16300  df-asp 16301  df-ascl 16302  df-psr 16345  df-mvr 16346  df-mpl 16347  df-evls 16348  df-evl 16349  df-opsr 16353  df-psr1 16504  df-vr1 16505  df-ply1 16506  df-evl1 16508  df-coe1 16509  df-cnfld 16628  df-mdeg 19846  df-deg1 19847  df-mon1 19921  df-uc1p 19922  df-q1p 19923  df-r1p 19924
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