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Theorem evlsval 19971
 Description: Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
Hypotheses
Ref Expression
evlsval.q evalSub
evlsval.w mPoly
evlsval.v mVar
evlsval.u s
evlsval.t s
evlsval.b
evlsval.a algSc
evlsval.x
evlsval.y
Assertion
Ref Expression
evlsval SubRing RingHom
Distinct variable groups:   ,,,   ,,   ,,,   ,   ,
Allowed substitution hints:   (,,)   (,,)   (,,)   ()   (,)   (,,)   (,,)   (,)   (,,)   (,,)

Proof of Theorem evlsval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlsval.q . . . 4 evalSub
2 fveq2 5757 . . . . . . . . 9
32adantl 454 . . . . . . . 8
43csbeq1d 3273 . . . . . . 7 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mVar s
5 fvex 5771 . . . . . . . . 9
65a1i 11 . . . . . . . 8
7 simplr 733 . . . . . . . . . 10
87fveq2d 5761 . . . . . . . . 9 SubRing SubRing
9 simpll 732 . . . . . . . . . . . 12
10 oveq1 6117 . . . . . . . . . . . . 13 s s
1110ad2antlr 709 . . . . . . . . . . . 12 s s
129, 11oveq12d 6128 . . . . . . . . . . 11 mPoly s mPoly s
1312csbeq1d 3273 . . . . . . . . . 10 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
14 ovex 6135 . . . . . . . . . . . 12 mPoly s
1514a1i 11 . . . . . . . . . . 11 mPoly s
16 simprr 735 . . . . . . . . . . . . . 14 mPoly s mPoly s
17 simplr 733 . . . . . . . . . . . . . . 15 mPoly s
18 simprl 734 . . . . . . . . . . . . . . . 16 mPoly s
19 simpll 732 . . . . . . . . . . . . . . . 16 mPoly s
2018, 19oveq12d 6128 . . . . . . . . . . . . . . 15 mPoly s
2117, 20oveq12d 6128 . . . . . . . . . . . . . 14 mPoly s s s
2216, 21oveq12d 6128 . . . . . . . . . . . . 13 mPoly s RingHom s mPoly s RingHom s
2316fveq2d 5761 . . . . . . . . . . . . . . . 16 mPoly s algSc algSc mPoly s
2423coeq2d 5064 . . . . . . . . . . . . . . 15 mPoly s algSc algSc mPoly s
2520xpeq1d 4930 . . . . . . . . . . . . . . . 16 mPoly s
2625mpteq2dv 4321 . . . . . . . . . . . . . . 15 mPoly s
2724, 26eqeq12d 2456 . . . . . . . . . . . . . 14 mPoly s algSc algSc mPoly s
2817oveq1d 6125 . . . . . . . . . . . . . . . . 17 mPoly s s s
2919, 28oveq12d 6128 . . . . . . . . . . . . . . . 16 mPoly s mVar s mVar s
3029coeq2d 5064 . . . . . . . . . . . . . . 15 mPoly s mVar s mVar s
3120mpteq1d 4315 . . . . . . . . . . . . . . . 16 mPoly s
3219, 31mpteq12dv 4312 . . . . . . . . . . . . . . 15 mPoly s
3330, 32eqeq12d 2456 . . . . . . . . . . . . . 14 mPoly s mVar s mVar s
3427, 33anbi12d 693 . . . . . . . . . . . . 13 mPoly s algSc mVar s algSc mPoly s mVar s
3522, 34riotaeqbidv 6581 . . . . . . . . . . . 12 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mPoly s mVar s
3635anassrs 631 . . . . . . . . . . 11 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mPoly s mVar s
3715, 36csbied 3292 . . . . . . . . . 10 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mPoly s mVar s
3813, 37eqtrd 2474 . . . . . . . . 9 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mPoly s mVar s
398, 38mpteq12dv 4312 . . . . . . . 8 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mPoly s mVar s
406, 39csbied 3292 . . . . . . 7 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mPoly s mVar s
414, 40eqtrd 2474 . . . . . 6 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mPoly s mVar s
42 df-evls 16451 . . . . . 6 evalSub SubRing mPoly s RingHom s algSc mVar s
43 fvex 5771 . . . . . . 7 SubRing
4443mptex 5995 . . . . . 6 SubRing mPoly s RingHom s algSc mPoly s mVar s
4541, 42, 44ovmpt2a 6233 . . . . 5 evalSub SubRing mPoly s RingHom s algSc mPoly s mVar s
4645fveq1d 5759 . . . 4 evalSub SubRing mPoly s RingHom s algSc mPoly s mVar s
471, 46syl5eq 2486 . . 3 SubRing mPoly s RingHom s algSc mPoly s mVar s
48473adant3 978 . 2 SubRing SubRing mPoly s RingHom s algSc mPoly s mVar s
49 oveq2 6118 . . . . . . . 8 s s
5049oveq2d 6126 . . . . . . 7 mPoly s mPoly s
5150oveq1d 6125 . . . . . 6 mPoly s RingHom s mPoly s RingHom s
5250fveq2d 5761 . . . . . . . . 9 algSc mPoly s algSc mPoly s
5352coeq2d 5064 . . . . . . . 8 algSc mPoly s algSc mPoly s
54 mpteq1 4314 . . . . . . . 8
5553, 54eqeq12d 2456 . . . . . . 7 algSc mPoly s algSc mPoly s
5649oveq2d 6126 . . . . . . . . 9 mVar s mVar s
5756coeq2d 5064 . . . . . . . 8 mVar s mVar s
5857eqeq1d 2450 . . . . . . 7 mVar s mVar s
5955, 58anbi12d 693 . . . . . 6 algSc mPoly s mVar s algSc mPoly s mVar s
6051, 59riotaeqbidv 6581 . . . . 5 mPoly s RingHom s algSc mPoly s mVar s mPoly s RingHom s algSc mPoly s mVar s
61 eqid 2442 . . . . 5 SubRing mPoly s RingHom s algSc mPoly s mVar s SubRing mPoly s RingHom s algSc mPoly s mVar s
62 riotaex 6582 . . . . 5 mPoly s RingHom s algSc mPoly s mVar s
6360, 61, 62fvmpt 5835 . . . 4 SubRing SubRing mPoly s RingHom s algSc mPoly s mVar s mPoly s RingHom s algSc mPoly s mVar s
64 evlsval.w . . . . . . . . 9 mPoly
65 evlsval.u . . . . . . . . . 10 s
6665oveq2i 6121 . . . . . . . . 9 mPoly mPoly s
6764, 66eqtri 2462 . . . . . . . 8 mPoly s
68 evlsval.t . . . . . . . . 9 s
69 evlsval.b . . . . . . . . . . 11
7069oveq1i 6120 . . . . . . . . . 10
7170oveq2i 6121 . . . . . . . . 9 s s
7268, 71eqtri 2462 . . . . . . . 8 s
7367, 72oveq12i 6122 . . . . . . 7 RingHom mPoly s RingHom s
7473a1i 11 . . . . . 6 RingHom mPoly s RingHom s
75 evlsval.a . . . . . . . . . . 11 algSc
7667fveq2i 5760 . . . . . . . . . . 11 algSc algSc mPoly s
7775, 76eqtri 2462 . . . . . . . . . 10 algSc mPoly s
7877coeq2i 5062 . . . . . . . . 9 algSc mPoly s
79 evlsval.x . . . . . . . . . 10
8070xpeq1i 4927 . . . . . . . . . . 11
8180mpteq2i 4317 . . . . . . . . . 10
8279, 81eqtri 2462 . . . . . . . . 9
8378, 82eqeq12i 2455 . . . . . . . 8 algSc mPoly s
84 evlsval.v . . . . . . . . . . 11 mVar
8565oveq2i 6121 . . . . . . . . . . 11 mVar mVar s
8684, 85eqtri 2462 . . . . . . . . . 10 mVar s
8786coeq2i 5062 . . . . . . . . 9 mVar s
88 evlsval.y . . . . . . . . . 10
89 eqid 2442 . . . . . . . . . . . 12
9070, 89mpteq12i 4318 . . . . . . . . . . 11
9190mpteq2i 4317 . . . . . . . . . 10
9288, 91eqtri 2462 . . . . . . . . 9
9387, 92eqeq12i 2455 . . . . . . . 8 mVar s
9483, 93anbi12i 680 . . . . . . 7 algSc mPoly s mVar s
9594a1i 11 . . . . . 6 algSc mPoly s mVar s
9674, 95riotaeqbidv 6581 . . . . 5 RingHom mPoly s RingHom s algSc mPoly s mVar s
9796trud 1333 . . . 4 RingHom mPoly s RingHom s algSc mPoly s mVar s
9863, 97syl6eqr 2492 . . 3 SubRing SubRing mPoly s RingHom s algSc mPoly s mVar s RingHom
99983ad2ant3 981 . 2 SubRing SubRing mPoly s RingHom s algSc mPoly s mVar s RingHom
10048, 99eqtrd 2474 1 SubRing RingHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wtru 1326   wceq 1653   wcel 1727  cvv 2962  csb 3267  csn 3838   cmpt 4291   cxp 4905   ccom 4911  cfv 5483  (class class class)co 6110  crio 6571   cmap 7047  cbs 13500   ↾s cress 13501   s cpws 13701  ccrg 15692   RingHom crh 15848  SubRingcsubrg 15895  algSccascl 16402   mVar cmvr 16438   mPoly cmpl 16439   evalSub ces 16440 This theorem is referenced by:  evlsval2  19972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-evls 16451
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