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Theorem pf1mpf 21124
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 21121 . 2 (𝐹𝑄𝑅 ∈ CRing)
3 id 22 . . . 4 (𝐹𝑄𝐹𝑄)
43, 1eleqtrdi 2843 . . 3 (𝐹𝑄𝐹 ∈ ran (eval1𝑅))
5 eqid 2738 . . . . . 6 (eval1𝑅) = (eval1𝑅)
6 eqid 2738 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
7 eqid 2738 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
8 pf1f.b . . . . . 6 𝐵 = (Base‘𝑅)
95, 6, 7, 8evl1rhm 21104 . . . . 5 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
102, 9syl 17 . . . 4 (𝐹𝑄 → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
11 eqid 2738 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2738 . . . . 5 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
1311, 12rhmf 19602 . . . 4 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
14 ffn 6504 . . . 4 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
15 fvelrnb 6732 . . . 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 235 . 2 (𝐹𝑄 → ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹)
18 eqid 2738 . . . . . . . 8 (1o eval 𝑅) = (1o eval 𝑅)
19 eqid 2738 . . . . . . . 8 (1o mPoly 𝑅) = (1o mPoly 𝑅)
20 eqid 2738 . . . . . . . . 9 (PwSer1𝑅) = (PwSer1𝑅)
216, 20, 11ply1bas 20972 . . . . . . . 8 (Base‘(Poly1𝑅)) = (Base‘(1o mPoly 𝑅))
225, 18, 8, 19, 21evl1val 21101 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1o × {𝑧}))))
2322coeq1d 5704 . . . . . 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 6098 . . . . . . 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 8145 . . . . . . . . . . 11 1o = {∅}
268fvexi 6690 . . . . . . . . . . 11 𝐵 ∈ V
27 0ex 5175 . . . . . . . . . . 11 ∅ ∈ V
28 eqid 2738 . . . . . . . . . . 11 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))
2925, 26, 27, 28mapsncnv 8505 . . . . . . . . . 10 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) = (𝑧𝐵 ↦ (1o × {𝑧}))
3029coeq1i 5702 . . . . . . . . 9 ((𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)))
3125, 26, 27, 28mapsnf1o2 8506 . . . . . . . . . 10 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)):(𝐵m 1o)–1-1-onto𝐵
32 f1ococnv1 6648 . . . . . . . . . 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 2787 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵m 1o)))
3534coeq2d 5705 . . . . . . 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, 35syl5eq 2785 . . . . . 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 2738 . . . . . . . 8 (𝑅s (𝐵m 1o)) = (𝑅s (𝐵m 1o))
38 eqid 2738 . . . . . . . 8 (Base‘(𝑅s (𝐵m 1o))) = (Base‘(𝑅s (𝐵m 1o)))
39 simpl 486 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → 𝑅 ∈ CRing)
40 ovexd 7207 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (𝐵m 1o) ∈ V)
41 1on 8140 . . . . . . . . . . 11 1o ∈ On
4218, 8, 19, 37evlrhm 20912 . . . . . . . . . . 11 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵m 1o))))
4341, 42mpan 690 . . . . . . . . . 10 (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵m 1o))))
4421, 38rhmf 19602 . . . . . . . . . 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))))
4645ffvelrnda 6863 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ (Base‘(𝑅s (𝐵m 1o))))
4737, 8, 38, 39, 40, 46pwselbas 16867 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵m 1o)⟶𝐵)
48 fcoi1 6552 . . . . . . 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 2777 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
5145ffnd 6505 . . . . . . 7 (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1𝑅)))
52 fnfvelrn 6860 . . . . . . 7 (((1o eval 𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
5351, 52sylan 583 . . . . . 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 2844 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
5650, 55eqeltrd 2833 . . . 4 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
57 coeq1 5700 . . . . 5 (((eval1𝑅)‘𝑦) = 𝐹 → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))))
5857eleq1d 2817 . . . 4 (((eval1𝑅)‘𝑦) = 𝐹 → ((((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
5956, 58syl5ibcom 248 . . 3 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
6059rexlimdva 3194 . 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 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  Vcvv 3398  c0 4211  {csn 4516  cmpt 5110   I cid 5428   × cxp 5523  ccnv 5524  ran crn 5526  cres 5527  ccom 5529  Oncon0 6172   Fn wfn 6334  wf 6335  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7172  1oc1o 8126  m cmap 8439  Basecbs 16588  s cpws 16825  CRingccrg 19419   RingHom crh 19588   mPoly cmpl 20721   eval cevl 20887  PwSer1cps1 20952  Poly1cpl1 20954  eval1ce1 21086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-cnex 10673  ax-resscn 10674  ax-1cn 10675  ax-icn 10676  ax-addcl 10677  ax-addrcl 10678  ax-mulcl 10679  ax-mulrcl 10680  ax-mulcom 10681  ax-addass 10682  ax-mulass 10683  ax-distr 10684  ax-i2m1 10685  ax-1ne0 10686  ax-1rid 10687  ax-rnegex 10688  ax-rrecex 10689  ax-cnre 10690  ax-pre-lttri 10691  ax-pre-lttrn 10692  ax-pre-ltadd 10693  ax-pre-mulgt0 10694
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-of 7427  df-ofr 7428  df-om 7602  df-1st 7716  df-2nd 7717  df-supp 7859  df-wrecs 7978  df-recs 8039  df-rdg 8077  df-1o 8133  df-er 8322  df-map 8441  df-pm 8442  df-ixp 8510  df-en 8558  df-dom 8559  df-sdom 8560  df-fin 8561  df-fsupp 8909  df-sup 8981  df-oi 9049  df-card 9443  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-sub 10952  df-neg 10953  df-nn 11719  df-2 11781  df-3 11782  df-4 11783  df-5 11784  df-6 11785  df-7 11786  df-8 11787  df-9 11788  df-n0 11979  df-z 12065  df-dec 12182  df-uz 12327  df-fz 12984  df-fzo 13127  df-seq 13463  df-hash 13785  df-struct 16590  df-ndx 16591  df-slot 16592  df-base 16594  df-sets 16595  df-ress 16596  df-plusg 16683  df-mulr 16684  df-sca 16686  df-vsca 16687  df-ip 16688  df-tset 16689  df-ple 16690  df-ds 16692  df-hom 16694  df-cco 16695  df-0g 16820  df-gsum 16821  df-prds 16826  df-pws 16828  df-mre 16962  df-mrc 16963  df-acs 16965  df-mgm 17970  df-sgrp 18019  df-mnd 18030  df-mhm 18074  df-submnd 18075  df-grp 18224  df-minusg 18225  df-sbg 18226  df-mulg 18345  df-subg 18396  df-ghm 18476  df-cntz 18567  df-cmn 19028  df-abl 19029  df-mgp 19361  df-ur 19373  df-srg 19377  df-ring 19420  df-cring 19421  df-rnghom 19591  df-subrg 19654  df-lmod 19757  df-lss 19825  df-lsp 19865  df-assa 20671  df-asp 20672  df-ascl 20673  df-psr 20724  df-mvr 20725  df-mpl 20726  df-opsr 20728  df-evls 20888  df-evl 20889  df-psr1 20957  df-ply1 20959  df-evl1 21088
This theorem is referenced by:  pf1ind  21127
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