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Theorem pf1mpf 19918
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 19915 . 2 (𝐹𝑄𝑅 ∈ CRing)
3 id 22 . . . 4 (𝐹𝑄𝐹𝑄)
43, 1syl6eleq 2849 . . 3 (𝐹𝑄𝐹 ∈ ran (eval1𝑅))
5 eqid 2760 . . . . . 6 (eval1𝑅) = (eval1𝑅)
6 eqid 2760 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
7 eqid 2760 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
8 pf1f.b . . . . . 6 𝐵 = (Base‘𝑅)
95, 6, 7, 8evl1rhm 19898 . . . . 5 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
102, 9syl 17 . . . 4 (𝐹𝑄 → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
11 eqid 2760 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2760 . . . . 5 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
1311, 12rhmf 18928 . . . 4 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
14 ffn 6206 . . . 4 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
15 fvelrnb 6405 . . . 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 2760 . . . . . . . 8 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
19 eqid 2760 . . . . . . . 8 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
20 eqid 2760 . . . . . . . . 9 (PwSer1𝑅) = (PwSer1𝑅)
216, 20, 11ply1bas 19767 . . . . . . . 8 (Base‘(Poly1𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
225, 18, 8, 19, 21evl1val 19895 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑦) = (((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1𝑜 × {𝑧}))))
2322coeq1d 5439 . . . . . 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 5815 . . . . . . 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 7741 . . . . . . . . . . 11 1𝑜 = {∅}
26 fvex 6362 . . . . . . . . . . . 12 (Base‘𝑅) ∈ V
278, 26eqeltri 2835 . . . . . . . . . . 11 𝐵 ∈ V
28 0ex 4942 . . . . . . . . . . 11 ∅ ∈ V
29 eqid 2760 . . . . . . . . . . 11 (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))
3025, 27, 28, 29mapsncnv 8070 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑧𝐵 ↦ (1𝑜 × {𝑧}))
3130coeq1i 5437 . . . . . . . . 9 ((𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ((𝑧𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)))
3225, 27, 28, 29mapsnf1o2 8071 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵
33 f1ococnv1 6326 . . . . . . . . . 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 2808 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑧𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵𝑚 1𝑜)))
3635coeq2d 5440 . . . . . . 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 2806 . . . . . 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 2760 . . . . . . . 8 (𝑅s (𝐵𝑚 1𝑜)) = (𝑅s (𝐵𝑚 1𝑜))
39 eqid 2760 . . . . . . . 8 (Base‘(𝑅s (𝐵𝑚 1𝑜))) = (Base‘(𝑅s (𝐵𝑚 1𝑜)))
40 simpl 474 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → 𝑅 ∈ CRing)
41 ovexd 6843 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (𝐵𝑚 1𝑜) ∈ V)
42 1on 7736 . . . . . . . . . . 11 1𝑜 ∈ On
4318, 8, 19, 38evlrhm 19727 . . . . . . . . . . 11 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))))
4442, 43mpan 708 . . . . . . . . . 10 (𝑅 ∈ CRing → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))))
4521, 39rhmf 18928 . . . . . . . . . 10 ((1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1𝑜))) → (1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))))
4644, 45syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → (1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))))
4746ffvelrnda 6522 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ (Base‘(𝑅s (𝐵𝑚 1𝑜))))
4838, 8, 39, 40, 41, 47pwselbas 16351 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦):(𝐵𝑚 1𝑜)⟶𝐵)
49 fcoi1 6239 . . . . . . 7 (((1𝑜 eval 𝑅)‘𝑦):(𝐵𝑚 1𝑜)⟶𝐵 → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
5048, 49syl 17 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
5123, 37, 503eqtrd 2798 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = ((1𝑜 eval 𝑅)‘𝑦))
52 ffn 6206 . . . . . . . 8 ((1𝑜 eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵𝑚 1𝑜))) → (1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)))
5346, 52syl 17 . . . . . . 7 (𝑅 ∈ CRing → (1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)))
54 fnfvelrn 6519 . . . . . . 7 (((1𝑜 eval 𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
5553, 54sylan 489 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
56 mpfpf1.q . . . . . 6 𝐸 = ran (1𝑜 eval 𝑅)
5755, 56syl6eleqr 2850 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ 𝐸)
5851, 57eqeltrd 2839 . . . 4 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
59 coeq1 5435 . . . . 5 (((eval1𝑅)‘𝑦) = 𝐹 → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))))
6059eleq1d 2824 . . . 4 (((eval1𝑅)‘𝑦) = 𝐹 → ((((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
6158, 60syl5ibcom 235 . . 3 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
6261rexlimdva 3169 . 2 (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
632, 17, 62sylc 65 1 (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  c0 4058  {csn 4321  cmpt 4881   I cid 5173   × cxp 5264  ccnv 5265  ran crn 5267  cres 5268  ccom 5270  Oncon0 5884   Fn wfn 6044  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  1𝑜c1o 7722  𝑚 cmap 8023  Basecbs 16059  s cpws 16309  CRingccrg 18748   RingHom crh 18914   mPoly cmpl 19555   eval cevl 19707  PwSer1cps1 19747  Poly1cpl1 19749  eval1ce1 19881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-gsum 16305  df-prds 16310  df-pws 16312  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-srg 18706  df-ring 18749  df-cring 18750  df-rnghom 18917  df-subrg 18980  df-lmod 19067  df-lss 19135  df-lsp 19174  df-assa 19514  df-asp 19515  df-ascl 19516  df-psr 19558  df-mvr 19559  df-mpl 19560  df-opsr 19562  df-evls 19708  df-evl 19709  df-psr1 19752  df-ply1 19754  df-evl1 19883
This theorem is referenced by:  pf1ind  19921
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