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Theorem mon1psubm 27628
Description: Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Hypotheses
Ref Expression
mon1psubm.p  |-  P  =  (Poly1 `  R )
mon1psubm.m  |-  M  =  (Monic1p `  R )
mon1psubm.u  |-  U  =  (mulGrp `  P )
Assertion
Ref Expression
mon1psubm  |-  ( R  e. NzRing  ->  M  e.  (SubMnd `  U ) )

Proof of Theorem mon1psubm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mon1psubm.p . . . . 5  |-  P  =  (Poly1 `  R )
2 eqid 2296 . . . . 5  |-  ( Base `  P )  =  (
Base `  P )
3 mon1psubm.m . . . . 5  |-  M  =  (Monic1p `  R )
41, 2, 3mon1pcl 19546 . . . 4  |-  ( x  e.  M  ->  x  e.  ( Base `  P
) )
54ssriv 3197 . . 3  |-  M  C_  ( Base `  P )
65a1i 10 . 2  |-  ( R  e. NzRing  ->  M  C_  ( Base `  P ) )
7 eqid 2296 . . . 4  |-  ( 1r
`  P )  =  ( 1r `  P
)
8 eqid 2296 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
91, 7, 3, 8mon1pid 27627 . . 3  |-  ( R  e. NzRing  ->  ( ( 1r
`  P )  e.  M  /\  ( ( deg1  `  R ) `  ( 1r `  P ) )  =  0 ) )
109simpld 445 . 2  |-  ( R  e. NzRing  ->  ( 1r `  P )  e.  M
)
111ply1nz 19523 . . . . . . 7  |-  ( R  e. NzRing  ->  P  e. NzRing )
12 nzrrng 16029 . . . . . . 7  |-  ( P  e. NzRing  ->  P  e.  Ring )
1311, 12syl 15 . . . . . 6  |-  ( R  e. NzRing  ->  P  e.  Ring )
1413adantr 451 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  P  e.  Ring )
154ad2antrl 708 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  x  e.  ( Base `  P )
)
16 simprr 733 . . . . . 6  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  y  e.  M )
175, 16sseldi 3191 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  y  e.  ( Base `  P )
)
18 eqid 2296 . . . . . 6  |-  ( .r
`  P )  =  ( .r `  P
)
192, 18rngcl 15370 . . . . 5  |-  ( ( P  e.  Ring  /\  x  e.  ( Base `  P
)  /\  y  e.  ( Base `  P )
)  ->  ( x
( .r `  P
) y )  e.  ( Base `  P
) )
2014, 15, 17, 19syl3anc 1182 . . . 4  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( x
( .r `  P
) y )  e.  ( Base `  P
) )
21 eqid 2296 . . . . . . 7  |-  (RLReg `  R )  =  (RLReg `  R )
22 eqid 2296 . . . . . . 7  |-  ( 0g
`  P )  =  ( 0g `  P
)
23 nzrrng 16029 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
2423adantr 451 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  R  e.  Ring )
251, 22, 3mon1pn0 19548 . . . . . . . 8  |-  ( x  e.  M  ->  x  =/=  ( 0g `  P
) )
2625ad2antrl 708 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  x  =/=  ( 0g `  P ) )
27 eqid 2296 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
288, 27, 3mon1pldg 19551 . . . . . . . . 9  |-  ( x  e.  M  ->  (
(coe1 `  x ) `  ( ( deg1  `  R ) `  x ) )  =  ( 1r `  R
) )
2928ad2antrl 708 . . . . . . . 8  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  x ) `  (
( deg1  `
 R ) `  x ) )  =  ( 1r `  R
) )
30 eqid 2296 . . . . . . . . . . . 12  |-  (Unit `  R )  =  (Unit `  R )
3121, 30unitrrg 16050 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
3223, 31syl 15 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  (Unit `  R
)  C_  (RLReg `  R
) )
3330, 271unit 15456 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  (Unit `  R )
)
3423, 33syl 15 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  ( 1r `  R )  e.  (Unit `  R ) )
3532, 34sseldd 3194 . . . . . . . . 9  |-  ( R  e. NzRing  ->  ( 1r `  R )  e.  (RLReg `  R ) )
3635adantr 451 . . . . . . . 8  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( 1r `  R )  e.  (RLReg `  R ) )
3729, 36eqeltrd 2370 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  x ) `  (
( deg1  `
 R ) `  x ) )  e.  (RLReg `  R )
)
381, 22, 3mon1pn0 19548 . . . . . . . 8  |-  ( y  e.  M  ->  y  =/=  ( 0g `  P
) )
3938ad2antll 709 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  y  =/=  ( 0g `  P ) )
408, 1, 21, 2, 18, 22, 24, 15, 26, 37, 17, 39deg1mul2 19516 . . . . . 6  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( ( deg1  `  R ) `  (
x ( .r `  P ) y ) )  =  ( ( ( deg1  `  R ) `  x )  +  ( ( deg1  `  R ) `  y ) ) )
418, 1, 22, 2deg1nn0cl 19490 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  P
)  /\  x  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  x
)  e.  NN0 )
4224, 15, 26, 41syl3anc 1182 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( ( deg1  `  R ) `  x
)  e.  NN0 )
438, 1, 22, 2deg1nn0cl 19490 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  P
)  /\  y  =/=  ( 0g `  P ) )  ->  ( ( deg1  `  R ) `  y
)  e.  NN0 )
4424, 17, 39, 43syl3anc 1182 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( ( deg1  `  R ) `  y
)  e.  NN0 )
4542, 44nn0addcld 10038 . . . . . 6  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (
( deg1  `
 R ) `  x )  +  ( ( deg1  `  R ) `  y ) )  e. 
NN0 )
4640, 45eqeltrd 2370 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( ( deg1  `  R ) `  (
x ( .r `  P ) y ) )  e.  NN0 )
478, 1, 22, 2deg1nn0clb 19492 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x ( .r `  P ) y )  e.  ( Base `  P
) )  ->  (
( x ( .r
`  P ) y )  =/=  ( 0g
`  P )  <->  ( ( deg1  `  R ) `  (
x ( .r `  P ) y ) )  e.  NN0 )
)
4824, 20, 47syl2anc 642 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (
x ( .r `  P ) y )  =/=  ( 0g `  P )  <->  ( ( deg1  `  R ) `  (
x ( .r `  P ) y ) )  e.  NN0 )
)
4946, 48mpbird 223 . . . 4  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( x
( .r `  P
) y )  =/=  ( 0g `  P
) )
5040fveq2d 5545 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  ( x ( .r
`  P ) y ) ) `  (
( deg1  `
 R ) `  ( x ( .r
`  P ) y ) ) )  =  ( (coe1 `  ( x ( .r `  P ) y ) ) `  ( ( ( deg1  `  R
) `  x )  +  ( ( deg1  `  R
) `  y )
) ) )
51 eqid 2296 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
521, 18, 51, 2, 8, 22, 24, 15, 26, 17, 39coe1mul4 19502 . . . . . 6  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  ( x ( .r
`  P ) y ) ) `  (
( ( deg1  `  R ) `  x )  +  ( ( deg1  `  R ) `  y ) ) )  =  ( ( (coe1 `  x ) `  (
( deg1  `
 R ) `  x ) ) ( .r `  R ) ( (coe1 `  y ) `  ( ( deg1  `  R ) `  y ) ) ) )
538, 27, 3mon1pldg 19551 . . . . . . . . 9  |-  ( y  e.  M  ->  (
(coe1 `  y ) `  ( ( deg1  `  R ) `  y ) )  =  ( 1r `  R
) )
5453ad2antll 709 . . . . . . . 8  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  y ) `  (
( deg1  `
 R ) `  y ) )  =  ( 1r `  R
) )
5529, 54oveq12d 5892 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (
(coe1 `  x ) `  ( ( deg1  `  R ) `  x ) ) ( .r `  R ) ( (coe1 `  y ) `  ( ( deg1  `  R ) `  y ) ) )  =  ( ( 1r
`  R ) ( .r `  R ) ( 1r `  R
) ) )
56 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
5756, 27rngidcl 15377 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
5823, 57syl 15 . . . . . . . . 9  |-  ( R  e. NzRing  ->  ( 1r `  R )  e.  (
Base `  R )
)
5956, 51, 27rnglidm 15380 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( 1r
`  R ) )  =  ( 1r `  R ) )
6023, 58, 59syl2anc 642 . . . . . . . 8  |-  ( R  e. NzRing  ->  ( ( 1r
`  R ) ( .r `  R ) ( 1r `  R
) )  =  ( 1r `  R ) )
6160adantr 451 . . . . . . 7  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( ( 1r `  R ) ( .r `  R ) ( 1r `  R
) )  =  ( 1r `  R ) )
6255, 61eqtrd 2328 . . . . . 6  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (
(coe1 `  x ) `  ( ( deg1  `  R ) `  x ) ) ( .r `  R ) ( (coe1 `  y ) `  ( ( deg1  `  R ) `  y ) ) )  =  ( 1r `  R ) )
6352, 62eqtrd 2328 . . . . 5  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  ( x ( .r
`  P ) y ) ) `  (
( ( deg1  `  R ) `  x )  +  ( ( deg1  `  R ) `  y ) ) )  =  ( 1r `  R ) )
6450, 63eqtrd 2328 . . . 4  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( (coe1 `  ( x ( .r
`  P ) y ) ) `  (
( deg1  `
 R ) `  ( x ( .r
`  P ) y ) ) )  =  ( 1r `  R
) )
651, 2, 22, 8, 3, 27ismon1p 19544 . . . 4  |-  ( ( x ( .r `  P ) y )  e.  M  <->  ( (
x ( .r `  P ) y )  e.  ( Base `  P
)  /\  ( x
( .r `  P
) y )  =/=  ( 0g `  P
)  /\  ( (coe1 `  ( x ( .r
`  P ) y ) ) `  (
( deg1  `
 R ) `  ( x ( .r
`  P ) y ) ) )  =  ( 1r `  R
) ) )
6620, 49, 64, 65syl3anbrc 1136 . . 3  |-  ( ( R  e. NzRing  /\  (
x  e.  M  /\  y  e.  M )
)  ->  ( x
( .r `  P
) y )  e.  M )
6766ralrimivva 2648 . 2  |-  ( R  e. NzRing  ->  A. x  e.  M  A. y  e.  M  ( x ( .r
`  P ) y )  e.  M )
68 mon1psubm.u . . . . 5  |-  U  =  (mulGrp `  P )
6968rngmgp 15363 . . . 4  |-  ( P  e.  Ring  ->  U  e. 
Mnd )
7013, 69syl 15 . . 3  |-  ( R  e. NzRing  ->  U  e.  Mnd )
7168, 2mgpbas 15347 . . . 4  |-  ( Base `  P )  =  (
Base `  U )
7268, 7rngidval 15359 . . . 4  |-  ( 1r
`  P )  =  ( 0g `  U
)
7368, 18mgpplusg 15345 . . . 4  |-  ( .r
`  P )  =  ( +g  `  U
)
7471, 72, 73issubm 14441 . . 3  |-  ( U  e.  Mnd  ->  ( M  e.  (SubMnd `  U
)  <->  ( M  C_  ( Base `  P )  /\  ( 1r `  P
)  e.  M  /\  A. x  e.  M  A. y  e.  M  (
x ( .r `  P ) y )  e.  M ) ) )
7570, 74syl 15 . 2  |-  ( R  e. NzRing  ->  ( M  e.  (SubMnd `  U )  <->  ( M  C_  ( Base `  P )  /\  ( 1r `  P )  e.  M  /\  A. x  e.  M  A. y  e.  M  ( x
( .r `  P
) y )  e.  M ) ) )
766, 10, 67, 75mpbir3and 1135 1  |-  ( R  e. NzRing  ->  M  e.  (SubMnd `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   0cc0 8753    + caddc 8756   NN0cn0 9981   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Mndcmnd 14377  SubMndcsubmnd 14430  mulGrpcmgp 15341   Ringcrg 15353   1rcur 15355  Unitcui 15437  NzRingcnzr 16025  RLRegcrlreg 16036  Poly1cpl1 16268  coe1cco1 16271   deg1 cdg1 19456  Monic1pcmn1 19527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-subrg 15559  df-lmod 15645  df-lss 15706  df-nzr 16026  df-rlreg 16040  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458  df-mon1 19532
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