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Theorem pf1mpf 19644
Description: Convert a univariate polynomial function to multivariate. (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
pf1mpf (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝑄   𝑥,𝑅
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem pf1mpf
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pf1rcl.q . . 3 𝑄 = ran (eval1𝑅)
21pf1rcl 19641 . 2 (𝐹𝑄𝑅 ∈ CRing)
3 id 22 . . . 4 (𝐹𝑄𝐹𝑄)
43, 1syl6eleq 2708 . . 3 (𝐹𝑄𝐹 ∈ ran (eval1𝑅))
5 eqid 2621 . . . . . 6 (eval1𝑅) = (eval1𝑅)
6 eqid 2621 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
7 eqid 2621 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
8 pf1f.b . . . . . 6 𝐵 = (Base‘𝑅)
95, 6, 7, 8evl1rhm 19624 . . . . 5 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
102, 9syl 17 . . . 4 (𝐹𝑄 → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
11 eqid 2621 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2621 . . . . 5 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
1311, 12rhmf 18654 . . . 4 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
14 ffn 6007 . . . 4 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
15 fvelrnb 6205 . . . 4 ((eval1𝑅) Fn (Base‘(Poly1𝑅)) → (𝐹 ∈ ran (eval1𝑅) ↔ ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹))
1610, 13, 14, 154syl 19 . . 3 (𝐹𝑄 → (𝐹 ∈ ran (eval1𝑅) ↔ ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹))
174, 16mpbid 222 . 2 (𝐹𝑄 → ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹)
18 eqid 2621 . . . . . . . 8 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
19 eqid 2621 . . . . . . . 8 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
20 eqid 2621 . . . . . . . . 9 (PwSer1𝑅) = (PwSer1𝑅)
216, 20, 11ply1bas 19493 . . . . . . . 8 (Base‘(Poly1𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
225, 18, 8, 19, 21evl1val 19621 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑦) = (((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1𝑜 × {𝑧}))))
2322coeq1d 5248 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ((((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1𝑜 × {𝑧}))) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))))
24 coass 5618 . . . . . . 7 ((((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1𝑜 × {𝑧}))) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = (((1𝑜 eval 𝑅)‘𝑦) ∘ ((𝑧𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))))
25 df1o2 7524 . . . . . . . . . . 11 1𝑜 = {∅}
26 fvex 6163 . . . . . . . . . . . 12 (Base‘𝑅) ∈ V
278, 26eqeltri 2694 . . . . . . . . . . 11 𝐵 ∈ V
28 0ex 4755 . . . . . . . . . . 11 ∅ ∈ V
29 eqid 2621 . . . . . . . . . . 11 (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))
3025, 27, 28, 29mapsncnv 7855 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑧𝐵 ↦ (1𝑜 × {𝑧}))
3130coeq1i 5246 . . . . . . . . 9 ((𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ((𝑧𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)))
3225, 27, 28, 29mapsnf1o2 7856 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵
33 f1ococnv1 6127 . . . . . . . . . 10 ((𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵 → ((𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵𝑚 1𝑜)))
3432, 33mp1i 13 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵𝑚 1𝑜)))
3531, 34syl5eqr 2669 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑧𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵𝑚 1𝑜)))
3635coeq2d 5249 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((1𝑜 eval 𝑅)‘𝑦) ∘ ((𝑧𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)))) = (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵𝑚 1𝑜))))
3724, 36syl5eq 2667 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1𝑜 × {𝑧}))) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵𝑚 1𝑜))))
38 eqid 2621 . . . . . . . 8 (𝑅s (𝐵𝑚 1𝑜)) = (𝑅s (𝐵𝑚 1𝑜))
39 eqid 2621 . . . . . . . 8 (Base‘(𝑅s (𝐵𝑚 1𝑜))) = (Base‘(𝑅s (𝐵𝑚 1𝑜)))
40 simpl 473 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → 𝑅 ∈ CRing)
41 ovex 6638 . . . . . . . . 9 (𝐵𝑚 1𝑜) ∈ V
4241a1i 11 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (𝐵𝑚 1𝑜) ∈ V)
43 1on 7519 . . . . . . . . . . 11 1𝑜 ∈ On
4418, 8, 19, 38evlrhm 19453 . . . . . . . . . . 11 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))))
4543, 44mpan 705 . . . . . . . . . 10 (𝑅 ∈ CRing → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))))
4621, 39rhmf 18654 . . . . . . . . . 10 ((1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))) → (1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))))
4745, 46syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → (1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))))
4847ffvelrnda 6320 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ (Base‘(𝑅s (𝐵𝑚 1𝑜))))
4938, 8, 39, 40, 42, 48pwselbas 16077 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦):(𝐵𝑚 1𝑜)⟶𝐵)
50 fcoi1 6040 . . . . . . 7 (((1𝑜 eval 𝑅)‘𝑦):(𝐵𝑚 1𝑜)⟶𝐵 → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
5149, 50syl 17 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
5223, 37, 513eqtrd 2659 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ((1𝑜 eval 𝑅)‘𝑦))
53 ffn 6007 . . . . . . . 8 ((1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))) → (1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)))
5447, 53syl 17 . . . . . . 7 (𝑅 ∈ CRing → (1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)))
55 fnfvelrn 6317 . . . . . . 7 (((1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
5654, 55sylan 488 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
57 mpfpf1.q . . . . . 6 𝐸 = ran (1𝑜 eval 𝑅)
5856, 57syl6eleqr 2709 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ 𝐸)
5952, 58eqeltrd 2698 . . . 4 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
60 coeq1 5244 . . . . 5 (((eval1𝑅)‘𝑦) = 𝐹 → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))))
6160eleq1d 2683 . . . 4 (((eval1𝑅)‘𝑦) = 𝐹 → ((((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
6259, 61syl5ibcom 235 . . 3 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
6362rexlimdva 3025 . 2 (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
642, 17, 63sylc 65 1 (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3189  c0 3896  {csn 4153  cmpt 4678   I cid 4989   × cxp 5077  ccnv 5078  ran crn 5080  cres 5081  ccom 5083  Oncon0 5687   Fn wfn 5847  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  1𝑜c1o 7505  𝑚 cmap 7809  Basecbs 15788  s cpws 16035  CRingccrg 18476   RingHom crh 18640   mPoly cmpl 19281   eval cevl 19433  PwSer1cps1 19473  Poly1cpl1 19475  eval1ce1 19607
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 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
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 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-sup 8299  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-fz 12276  df-fzo 12414  df-seq 12749  df-hash 13065  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-hom 15894  df-cco 15895  df-0g 16030  df-gsum 16031  df-prds 16036  df-pws 16038  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-mhm 17263  df-submnd 17264  df-grp 17353  df-minusg 17354  df-sbg 17355  df-mulg 17469  df-subg 17519  df-ghm 17586  df-cntz 17678  df-cmn 18123  df-abl 18124  df-mgp 18418  df-ur 18430  df-srg 18434  df-ring 18477  df-cring 18478  df-rnghom 18643  df-subrg 18706  df-lmod 18793  df-lss 18861  df-lsp 18900  df-assa 19240  df-asp 19241  df-ascl 19242  df-psr 19284  df-mvr 19285  df-mpl 19286  df-opsr 19288  df-evls 19434  df-evl 19435  df-psr1 19478  df-ply1 19480  df-evl1 19609
This theorem is referenced by:  pf1ind  19647
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