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Theorem mpfrcl 19892
 Description: Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Hypothesis
Ref Expression
mpfrcl.q evalSub
Assertion
Ref Expression
mpfrcl SubRing

Proof of Theorem mpfrcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3594 . . 3 evalSub evalSub
2 mpfrcl.q . . 3 evalSub
31, 2eleq2s 2496 . 2 evalSub
4 rneq 5054 . . . 4 evalSub evalSub
5 rn0 5086 . . . 4
64, 5syl6eq 2452 . . 3 evalSub evalSub
76necon3i 2606 . 2 evalSub evalSub
8 fveq1 5686 . . . . . . 7 evalSub evalSub
9 fv01 5722 . . . . . . 7
108, 9syl6eq 2452 . . . . . 6 evalSub evalSub
1110necon3i 2606 . . . . 5 evalSub evalSub
12 reldmevls 19891 . . . . . . . 8 evalSub
1312ovprc1 6068 . . . . . . 7 evalSub
1413necon1ai 2609 . . . . . 6 evalSub
15 n0 3597 . . . . . . 7 evalSub evalSub
16 df-evls 16375 . . . . . . . . . 10 evalSub SubRing mPoly s RingHom s algSc mVar s
1716elmpt2cl2 6249 . . . . . . . . 9 evalSub
1817a1d 23 . . . . . . . 8 evalSub
1918exlimiv 1641 . . . . . . 7 evalSub
2015, 19sylbi 188 . . . . . 6 evalSub
2114, 20jcai 523 . . . . 5 evalSub
2211, 21syl 16 . . . 4 evalSub
23 fvex 5701 . . . . . . . . . . . . 13
24 nfcv 2540 . . . . . . . . . . . . . 14 SubRing
25 nfcsb1v 3243 . . . . . . . . . . . . . 14 mPoly s RingHom s algSc mVar s
2624, 25nfmpt 4257 . . . . . . . . . . . . 13 SubRing mPoly s RingHom s algSc mVar s
27 csbeq1a 3219 . . . . . . . . . . . . . 14 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
2827mpteq2dv 4256 . . . . . . . . . . . . 13 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mVar s
2923, 26, 28csbief 3252 . . . . . . . . . . . 12 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mVar s
30 fveq2 5687 . . . . . . . . . . . . . 14 SubRing SubRing
3130adantl 453 . . . . . . . . . . . . 13 SubRing SubRing
32 fveq2 5687 . . . . . . . . . . . . . . . 16
3332adantl 453 . . . . . . . . . . . . . . 15
3433csbeq1d 3217 . . . . . . . . . . . . . 14 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
35 id 20 . . . . . . . . . . . . . . . . . 18
36 oveq1 6047 . . . . . . . . . . . . . . . . . 18 s s
3735, 36oveqan12d 6059 . . . . . . . . . . . . . . . . 17 mPoly s mPoly s
3837csbeq1d 3217 . . . . . . . . . . . . . . . 16 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
39 id 20 . . . . . . . . . . . . . . . . . . . 20
40 oveq2 6048 . . . . . . . . . . . . . . . . . . . 20
4139, 40oveqan12rd 6060 . . . . . . . . . . . . . . . . . . 19 s s
4241oveq2d 6056 . . . . . . . . . . . . . . . . . 18 RingHom s RingHom s
4340adantr 452 . . . . . . . . . . . . . . . . . . . . . 22
4443xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . 21
4544mpteq2dv 4256 . . . . . . . . . . . . . . . . . . . 20
4645eqeq2d 2415 . . . . . . . . . . . . . . . . . . 19 algSc algSc
4735, 36oveqan12d 6059 . . . . . . . . . . . . . . . . . . . . 21 mVar s mVar s
4847coeq2d 4994 . . . . . . . . . . . . . . . . . . . 20 mVar s mVar s
49 simpl 444 . . . . . . . . . . . . . . . . . . . . 21
5043mpteq1d 4250 . . . . . . . . . . . . . . . . . . . . 21
5149, 50mpteq12dv 4247 . . . . . . . . . . . . . . . . . . . 20
5248, 51eqeq12d 2418 . . . . . . . . . . . . . . . . . . 19 mVar s mVar s
5346, 52anbi12d 692 . . . . . . . . . . . . . . . . . 18 algSc mVar s algSc mVar s
5442, 53riotaeqbidv 6511 . . . . . . . . . . . . . . . . 17 RingHom s algSc mVar s RingHom s algSc mVar s
5554csbeq2dv 3236 . . . . . . . . . . . . . . . 16 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
5638, 55eqtrd 2436 . . . . . . . . . . . . . . 15 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
5756csbeq2dv 3236 . . . . . . . . . . . . . 14 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
5834, 57eqtrd 2436 . . . . . . . . . . . . 13 mPoly s RingHom s algSc mVar s mPoly s RingHom s algSc mVar s
5931, 58mpteq12dv 4247 . . . . . . . . . . . 12 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mVar s
6029, 59syl5eq 2448 . . . . . . . . . . 11 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mVar s
61 fvex 5701 . . . . . . . . . . . 12 SubRing
6261mptex 5925 . . . . . . . . . . 11 SubRing mPoly s RingHom s algSc mVar s
6360, 16, 62ovmpt2a 6163 . . . . . . . . . 10 evalSub SubRing mPoly s RingHom s algSc mVar s
6463dmeqd 5031 . . . . . . . . 9 evalSub SubRing mPoly s RingHom s algSc mVar s
65 eqid 2404 . . . . . . . . . 10 SubRing mPoly s RingHom s algSc mVar s SubRing mPoly s RingHom s algSc mVar s
6665dmmptss 5325 . . . . . . . . 9 SubRing mPoly s RingHom s algSc mVar s SubRing
6764, 66syl6eqss 3358 . . . . . . . 8 evalSub SubRing
6867ssneld 3310 . . . . . . 7 SubRing evalSub
69 ndmfv 5714 . . . . . . 7 evalSub evalSub
7068, 69syl6 31 . . . . . 6 SubRing evalSub
7170necon1ad 2634 . . . . 5 evalSub SubRing
7271com12 29 . . . 4 evalSub SubRing
7322, 72jcai 523 . . 3 evalSub SubRing
74 df-3an 938 . . 3 SubRing SubRing
7573, 74sylibr 204 . 2 evalSub SubRing
763, 7, 753syl 19 1 SubRing
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   w3a 936  wex 1547   wceq 1649   wcel 1721   wne 2567  cvv 2916  csb 3211  c0 3588  csn 3774   cmpt 4226   cxp 4835   cdm 4837   crn 4838   ccom 4841  cfv 5413  (class class class)co 6040  crio 6501   cmap 6977  cbs 13424   ↾s cress 13425   s cpws 13625  ccrg 15616   RingHom crh 15772  SubRingcsubrg 15819  algSccascl 16326   mVar cmvr 16362   mPoly cmpl 16363   evalSub ces 16364 This theorem is referenced by:  mpff  19915  mpfaddcl  19916  mpfmulcl  19917  mpfind  19918  pf1rcl  19922  mpfpf1  19924 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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-evls 16375
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