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Theorem evl1addd 19521
Description: Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
Hypotheses
Ref Expression
evl1addd.q  |-  O  =  (eval1 `  R )
evl1addd.p  |-  P  =  (Poly1 `  R )
evl1addd.b  |-  B  =  ( Base `  R
)
evl1addd.u  |-  U  =  ( Base `  P
)
evl1addd.1  |-  ( ph  ->  R  e.  CRing )
evl1addd.2  |-  ( ph  ->  Y  e.  B )
evl1addd.3  |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y
)  =  V ) )
evl1addd.4  |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y
)  =  W ) )
evl1addd.g  |-  .+b  =  ( +g  `  P )
evl1addd.a  |-  .+  =  ( +g  `  R )
Assertion
Ref Expression
evl1addd  |-  ( ph  ->  ( ( M  .+b  N )  e.  U  /\  ( ( O `  ( M  .+b  N ) ) `  Y )  =  ( V  .+  W ) ) )

Proof of Theorem evl1addd
StepHypRef Expression
1 evl1addd.1 . . . . . 6  |-  ( ph  ->  R  e.  CRing )
2 evl1addd.q . . . . . . 7  |-  O  =  (eval1 `  R )
3 evl1addd.p . . . . . . 7  |-  P  =  (Poly1 `  R )
4 eqid 2358 . . . . . . 7  |-  ( R  ^s  B )  =  ( R  ^s  B )
5 evl1addd.b . . . . . . 7  |-  B  =  ( Base `  R
)
62, 3, 4, 5evl1rhm 19516 . . . . . 6  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  B
) ) )
71, 6syl 15 . . . . 5  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  B ) ) )
8 rhmghm 15602 . . . . 5  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O  e.  ( P  GrpHom  ( R  ^s  B )
) )
97, 8syl 15 . . . 4  |-  ( ph  ->  O  e.  ( P 
GrpHom  ( R  ^s  B ) ) )
10 ghmgrp1 14784 . . . 4  |-  ( O  e.  ( P  GrpHom  ( R  ^s  B ) )  ->  P  e.  Grp )
119, 10syl 15 . . 3  |-  ( ph  ->  P  e.  Grp )
12 evl1addd.3 . . . 4  |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y
)  =  V ) )
1312simpld 445 . . 3  |-  ( ph  ->  M  e.  U )
14 evl1addd.4 . . . 4  |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y
)  =  W ) )
1514simpld 445 . . 3  |-  ( ph  ->  N  e.  U )
16 evl1addd.u . . . 4  |-  U  =  ( Base `  P
)
17 evl1addd.g . . . 4  |-  .+b  =  ( +g  `  P )
1816, 17grpcl 14594 . . 3  |-  ( ( P  e.  Grp  /\  M  e.  U  /\  N  e.  U )  ->  ( M  .+b  N
)  e.  U )
1911, 13, 15, 18syl3anc 1182 . 2  |-  ( ph  ->  ( M  .+b  N
)  e.  U )
20 eqid 2358 . . . . . . 7  |-  ( +g  `  ( R  ^s  B ) )  =  ( +g  `  ( R  ^s  B ) )
2116, 17, 20ghmlin 14787 . . . . . 6  |-  ( ( O  e.  ( P 
GrpHom  ( R  ^s  B ) )  /\  M  e.  U  /\  N  e.  U )  ->  ( O `  ( M  .+b 
N ) )  =  ( ( O `  M ) ( +g  `  ( R  ^s  B ) ) ( O `  N ) ) )
229, 13, 15, 21syl3anc 1182 . . . . 5  |-  ( ph  ->  ( O `  ( M  .+b  N ) )  =  ( ( O `
 M ) ( +g  `  ( R  ^s  B ) ) ( O `  N ) ) )
23 eqid 2358 . . . . . 6  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
24 fvex 5622 . . . . . . . 8  |-  ( Base `  R )  e.  _V
255, 24eqeltri 2428 . . . . . . 7  |-  B  e. 
_V
2625a1i 10 . . . . . 6  |-  ( ph  ->  B  e.  _V )
2716, 23rhmf 15603 . . . . . . . 8  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
287, 27syl 15 . . . . . . 7  |-  ( ph  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
29 ffvelrn 5746 . . . . . . 7  |-  ( ( O : U --> ( Base `  ( R  ^s  B ) )  /\  M  e.  U )  ->  ( O `  M )  e.  ( Base `  ( R  ^s  B ) ) )
3028, 13, 29syl2anc 642 . . . . . 6  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( R  ^s  B ) ) )
31 ffvelrn 5746 . . . . . . 7  |-  ( ( O : U --> ( Base `  ( R  ^s  B ) )  /\  N  e.  U )  ->  ( O `  N )  e.  ( Base `  ( R  ^s  B ) ) )
3228, 15, 31syl2anc 642 . . . . . 6  |-  ( ph  ->  ( O `  N
)  e.  ( Base `  ( R  ^s  B ) ) )
33 evl1addd.a . . . . . 6  |-  .+  =  ( +g  `  R )
344, 23, 1, 26, 30, 32, 33, 20pwsplusgval 13488 . . . . 5  |-  ( ph  ->  ( ( O `  M ) ( +g  `  ( R  ^s  B ) ) ( O `  N ) )  =  ( ( O `  M )  o F 
.+  ( O `  N ) ) )
3522, 34eqtrd 2390 . . . 4  |-  ( ph  ->  ( O `  ( M  .+b  N ) )  =  ( ( O `
 M )  o F  .+  ( O `
 N ) ) )
3635fveq1d 5610 . . 3  |-  ( ph  ->  ( ( O `  ( M  .+b  N ) ) `  Y )  =  ( ( ( O `  M )  o F  .+  ( O `  N )
) `  Y )
)
374, 5, 23, 1, 26, 30pwselbas 13487 . . . . 5  |-  ( ph  ->  ( O `  M
) : B --> B )
38 ffn 5472 . . . . 5  |-  ( ( O `  M ) : B --> B  -> 
( O `  M
)  Fn  B )
3937, 38syl 15 . . . 4  |-  ( ph  ->  ( O `  M
)  Fn  B )
404, 5, 23, 1, 26, 32pwselbas 13487 . . . . 5  |-  ( ph  ->  ( O `  N
) : B --> B )
41 ffn 5472 . . . . 5  |-  ( ( O `  N ) : B --> B  -> 
( O `  N
)  Fn  B )
4240, 41syl 15 . . . 4  |-  ( ph  ->  ( O `  N
)  Fn  B )
43 evl1addd.2 . . . 4  |-  ( ph  ->  Y  e.  B )
44 fnfvof 6177 . . . 4  |-  ( ( ( ( O `  M )  Fn  B  /\  ( O `  N
)  Fn  B )  /\  ( B  e. 
_V  /\  Y  e.  B ) )  -> 
( ( ( O `
 M )  o F  .+  ( O `
 N ) ) `
 Y )  =  ( ( ( O `
 M ) `  Y )  .+  (
( O `  N
) `  Y )
) )
4539, 42, 26, 43, 44syl22anc 1183 . . 3  |-  ( ph  ->  ( ( ( O `
 M )  o F  .+  ( O `
 N ) ) `
 Y )  =  ( ( ( O `
 M ) `  Y )  .+  (
( O `  N
) `  Y )
) )
4612simprd 449 . . . 4  |-  ( ph  ->  ( ( O `  M ) `  Y
)  =  V )
4714simprd 449 . . . 4  |-  ( ph  ->  ( ( O `  N ) `  Y
)  =  W )
4846, 47oveq12d 5963 . . 3  |-  ( ph  ->  ( ( ( O `
 M ) `  Y )  .+  (
( O `  N
) `  Y )
)  =  ( V 
.+  W ) )
4936, 45, 483eqtrd 2394 . 2  |-  ( ph  ->  ( ( O `  ( M  .+b  N ) ) `  Y )  =  ( V  .+  W ) )
5019, 49jca 518 1  |-  ( ph  ->  ( ( M  .+b  N )  e.  U  /\  ( ( O `  ( M  .+b  N ) ) `  Y )  =  ( V  .+  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Fcof 6163   Basecbs 13245   +g cplusg 13305    ^s cpws 13446   Grpcgrp 14461    GrpHom cghm 14779   CRingccrg 15437   RingHom crh 15593  Poly1cpl1 16351  eval1ce1 16353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-hom 13329  df-cco 13330  df-prds 13447  df-pws 13449  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-cring 15440  df-ur 15441  df-rnghom 15595  df-subrg 15642  df-lmod 15728  df-lss 15789  df-lsp 15828  df-assa 16152  df-asp 16153  df-ascl 16154  df-psr 16197  df-mvr 16198  df-mpl 16199  df-evls 16200  df-evl 16201  df-opsr 16205  df-psr1 16356  df-ply1 16358  df-evl1 16360
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