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Theorem mpfpf1 19647
 Description: Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pf1rcl.q 𝑄 = ran (eval1𝑅)
pf1f.b 𝐵 = (Base‘𝑅)
mpfpf1.q 𝐸 = ran (1𝑜 eval 𝑅)
Assertion
Ref Expression
mpfpf1 (𝐹𝐸 → (𝐹 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄)
Distinct variable groups:   𝑦,𝐵   𝑦,𝐸   𝑦,𝐹   𝑦,𝑅
Allowed substitution hint:   𝑄(𝑦)

Proof of Theorem mpfpf1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mpfpf1.q . . . . 5 𝐸 = ran (1𝑜 eval 𝑅)
2 eqid 2621 . . . . . . 7 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
3 pf1f.b . . . . . . 7 𝐵 = (Base‘𝑅)
42, 3evlval 19456 . . . . . 6 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
54rneqi 5317 . . . . 5 ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘𝐵)
61, 5eqtri 2643 . . . 4 𝐸 = ran ((1𝑜 evalSub 𝑅)‘𝐵)
76mpfrcl 19450 . . 3 (𝐹𝐸 → (1𝑜 ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅)))
87simp2d 1072 . 2 (𝐹𝐸𝑅 ∈ CRing)
9 id 22 . . . 4 (𝐹𝐸𝐹𝐸)
109, 1syl6eleq 2708 . . 3 (𝐹𝐸𝐹 ∈ ran (1𝑜 eval 𝑅))
11 1on 7519 . . . . 5 1𝑜 ∈ On
12 eqid 2621 . . . . . 6 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
13 eqid 2621 . . . . . 6 (𝑅s (𝐵𝑚 1𝑜)) = (𝑅s (𝐵𝑚 1𝑜))
142, 3, 12, 13evlrhm 19457 . . . . 5 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))))
1511, 8, 14sylancr 694 . . . 4 (𝐹𝐸 → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))))
16 eqid 2621 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
17 eqid 2621 . . . . . 6 (PwSer1𝑅) = (PwSer1𝑅)
18 eqid 2621 . . . . . 6 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
1916, 17, 18ply1bas 19497 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
20 eqid 2621 . . . . 5 (Base‘(𝑅s (𝐵𝑚 1𝑜))) = (Base‘(𝑅s (𝐵𝑚 1𝑜)))
2119, 20rhmf 18658 . . . 4 ((1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))) → (1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))))
22 ffn 6007 . . . 4 ((1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))) → (1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)))
23 fvelrnb 6205 . . . 4 ((1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)) → (𝐹 ∈ ran (1𝑜 eval 𝑅) ↔ ∃𝑥 ∈ (Base‘(Poly1𝑅))((1𝑜 eval 𝑅)‘𝑥) = 𝐹))
2415, 21, 22, 234syl 19 . . 3 (𝐹𝐸 → (𝐹 ∈ ran (1𝑜 eval 𝑅) ↔ ∃𝑥 ∈ (Base‘(Poly1𝑅))((1𝑜 eval 𝑅)‘𝑥) = 𝐹))
2510, 24mpbid 222 . 2 (𝐹𝐸 → ∃𝑥 ∈ (Base‘(Poly1𝑅))((1𝑜 eval 𝑅)‘𝑥) = 𝐹)
26 eqid 2621 . . . . . 6 (eval1𝑅) = (eval1𝑅)
2726, 2, 3, 12, 19evl1val 19625 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑥) = (((1𝑜 eval 𝑅)‘𝑥) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
28 eqid 2621 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
2926, 16, 28, 3evl1rhm 19628 . . . . . . . 8 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
30 eqid 2621 . . . . . . . . 9 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
3118, 30rhmf 18658 . . . . . . . 8 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
32 ffn 6007 . . . . . . . 8 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
3329, 31, 323syl 18 . . . . . . 7 (𝑅 ∈ CRing → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
34 fnfvelrn 6317 . . . . . . 7 (((eval1𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑥 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑥) ∈ ran (eval1𝑅))
3533, 34sylan 488 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑥) ∈ ran (eval1𝑅))
36 pf1rcl.q . . . . . 6 𝑄 = ran (eval1𝑅)
3735, 36syl6eleqr 2709 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑥) ∈ 𝑄)
3827, 37eqeltrrd 2699 . . . 4 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1𝑅))) → (((1𝑜 eval 𝑅)‘𝑥) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄)
39 coeq1 5244 . . . . 5 (((1𝑜 eval 𝑅)‘𝑥) = 𝐹 → (((1𝑜 eval 𝑅)‘𝑥) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) = (𝐹 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
4039eleq1d 2683 . . . 4 (((1𝑜 eval 𝑅)‘𝑥) = 𝐹 → ((((1𝑜 eval 𝑅)‘𝑥) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄 ↔ (𝐹 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄))
4138, 40syl5ibcom 235 . . 3 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly1𝑅))) → (((1𝑜 eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄))
4241rexlimdva 3025 . 2 (𝑅 ∈ CRing → (∃𝑥 ∈ (Base‘(Poly1𝑅))((1𝑜 eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄))
438, 25, 42sylc 65 1 (𝐹𝐸 → (𝐹 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))) ∈ 𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  Vcvv 3189  {csn 4153   ↦ cmpt 4678   × cxp 5077  ran crn 5080   ∘ ccom 5083  Oncon0 5687   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610  1𝑜c1o 7505   ↑𝑚 cmap 7809  Basecbs 15792   ↑s cpws 16039  CRingccrg 18480   RingHom crh 18644  SubRingcsubrg 18708   mPoly cmpl 19285   evalSub ces 19436   eval cevl 19437  PwSer1cps1 19477  Poly1cpl1 19479  eval1ce1 19611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-hom 15898  df-cco 15899  df-0g 16034  df-gsum 16035  df-prds 16040  df-pws 16042  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-cntz 17682  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-srg 18438  df-ring 18481  df-cring 18482  df-rnghom 18647  df-subrg 18710  df-lmod 18797  df-lss 18865  df-lsp 18904  df-assa 19244  df-asp 19245  df-ascl 19246  df-psr 19288  df-mvr 19289  df-mpl 19290  df-opsr 19292  df-evls 19438  df-evl 19439  df-psr1 19482  df-ply1 19484  df-evl1 19613 This theorem is referenced by:  pf1ind  19651
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