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Theorem pf1mpf 21718
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 21715 . 2 (𝐹𝑄𝑅 ∈ CRing)
3 id 22 . . . 4 (𝐹𝑄𝐹𝑄)
43, 1eleqtrdi 2848 . . 3 (𝐹𝑄𝐹 ∈ ran (eval1𝑅))
5 eqid 2736 . . . . . 6 (eval1𝑅) = (eval1𝑅)
6 eqid 2736 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
7 eqid 2736 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
8 pf1f.b . . . . . 6 𝐵 = (Base‘𝑅)
95, 6, 7, 8evl1rhm 21698 . . . . 5 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
102, 9syl 17 . . . 4 (𝐹𝑄 → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
11 eqid 2736 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2736 . . . . 5 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
1311, 12rhmf 20158 . . . 4 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
14 ffn 6668 . . . 4 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
15 fvelrnb 6903 . . . 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 231 . 2 (𝐹𝑄 → ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹)
18 eqid 2736 . . . . . . . 8 (1o eval 𝑅) = (1o eval 𝑅)
19 eqid 2736 . . . . . . . 8 (1o mPoly 𝑅) = (1o mPoly 𝑅)
20 eqid 2736 . . . . . . . . 9 (PwSer1𝑅) = (PwSer1𝑅)
216, 20, 11ply1bas 21566 . . . . . . . 8 (Base‘(Poly1𝑅)) = (Base‘(1o mPoly 𝑅))
225, 18, 8, 19, 21evl1val 21695 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1o × {𝑧}))))
2322coeq1d 5817 . . . . . 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 6217 . . . . . . 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 8419 . . . . . . . . . . 11 1o = {∅}
268fvexi 6856 . . . . . . . . . . 11 𝐵 ∈ V
27 0ex 5264 . . . . . . . . . . 11 ∅ ∈ V
28 eqid 2736 . . . . . . . . . . 11 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))
2925, 26, 27, 28mapsncnv 8831 . . . . . . . . . 10 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) = (𝑧𝐵 ↦ (1o × {𝑧}))
3029coeq1i 5815 . . . . . . . . 9 ((𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)))
3125, 26, 27, 28mapsnf1o2 8832 . . . . . . . . . 10 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)):(𝐵m 1o)–1-1-onto𝐵
32 f1ococnv1 6813 . . . . . . . . . 10 ((𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)):(𝐵m 1o)–1-1-onto𝐵 → ((𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵m 1o)))
3331, 32mp1i 13 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵m 1o)))
3430, 33eqtr3id 2790 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵m 1o)))
3534coeq2d 5818 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((1o eval 𝑅)‘𝑦) ∘ ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)))) = (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵m 1o))))
3624, 35eqtrid 2788 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((((1o eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1o × {𝑧}))) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵m 1o))))
37 eqid 2736 . . . . . . . 8 (𝑅s (𝐵m 1o)) = (𝑅s (𝐵m 1o))
38 eqid 2736 . . . . . . . 8 (Base‘(𝑅s (𝐵m 1o))) = (Base‘(𝑅s (𝐵m 1o)))
39 simpl 483 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → 𝑅 ∈ CRing)
40 ovexd 7392 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (𝐵m 1o) ∈ V)
41 1on 8424 . . . . . . . . . . 11 1o ∈ On
4218, 8, 19, 37evlrhm 21506 . . . . . . . . . . 11 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵m 1o))))
4341, 42mpan 688 . . . . . . . . . 10 (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵m 1o))))
4421, 38rhmf 20158 . . . . . . . . . 10 ((1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵m 1o))) → (1o eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵m 1o))))
4543, 44syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → (1o eval 𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s (𝐵m 1o))))
4645ffvelcdmda 7035 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ (Base‘(𝑅s (𝐵m 1o))))
4737, 8, 38, 39, 40, 46pwselbas 17371 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵m 1o)⟶𝐵)
48 fcoi1 6716 . . . . . . 7 (((1o eval 𝑅)‘𝑦):(𝐵m 1o)⟶𝐵 → (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵m 1o))) = ((1o eval 𝑅)‘𝑦))
4947, 48syl 17 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵m 1o))) = ((1o eval 𝑅)‘𝑦))
5023, 36, 493eqtrd 2780 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
5145ffnd 6669 . . . . . . 7 (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1𝑅)))
52 fnfvelrn 7031 . . . . . . 7 (((1o eval 𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
5351, 52sylan 580 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
54 mpfpf1.q . . . . . 6 𝐸 = ran (1o eval 𝑅)
5553, 54eleqtrrdi 2849 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
5650, 55eqeltrd 2838 . . . 4 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
57 coeq1 5813 . . . . 5 (((eval1𝑅)‘𝑦) = 𝐹 → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))))
5857eleq1d 2822 . . . 4 (((eval1𝑅)‘𝑦) = 𝐹 → ((((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
5956, 58syl5ibcom 244 . . 3 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
6059rexlimdva 3152 . 2 (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
612, 17, 60sylc 65 1 (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  c0 4282  {csn 4586  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  ccom 5637  Oncon0 6317   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  1oc1o 8405  m cmap 8765  Basecbs 17083  s cpws 17328  CRingccrg 19965   RingHom crh 20143   mPoly cmpl 21308   eval cevl 21481  PwSer1cps1 21546  Poly1cpl1 21548  eval1ce1 21680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-rnghom 20146  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-assa 21259  df-asp 21260  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-evls 21482  df-evl 21483  df-psr1 21551  df-ply1 21553  df-evl1 21682
This theorem is referenced by:  pf1ind  21721
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