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Theorem pf1mpf 20076
 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 20073 . 2 (𝐹𝑄𝑅 ∈ CRing)
3 id 22 . . . 4 (𝐹𝑄𝐹𝑄)
43, 1syl6eleq 2916 . . 3 (𝐹𝑄𝐹 ∈ ran (eval1𝑅))
5 eqid 2825 . . . . . 6 (eval1𝑅) = (eval1𝑅)
6 eqid 2825 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
7 eqid 2825 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
8 pf1f.b . . . . . 6 𝐵 = (Base‘𝑅)
95, 6, 7, 8evl1rhm 20056 . . . . 5 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
102, 9syl 17 . . . 4 (𝐹𝑄 → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
11 eqid 2825 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2825 . . . . 5 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
1311, 12rhmf 19082 . . . 4 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
14 ffn 6278 . . . 4 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
15 fvelrnb 6490 . . . 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 224 . 2 (𝐹𝑄 → ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹)
18 eqid 2825 . . . . . . . 8 (1o eval 𝑅) = (1o eval 𝑅)
19 eqid 2825 . . . . . . . 8 (1o mPoly 𝑅) = (1o mPoly 𝑅)
20 eqid 2825 . . . . . . . . 9 (PwSer1𝑅) = (PwSer1𝑅)
216, 20, 11ply1bas 19925 . . . . . . . 8 (Base‘(Poly1𝑅)) = (Base‘(1o mPoly 𝑅))
225, 18, 8, 19, 21evl1val 20053 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1o × {𝑧}))))
2322coeq1d 5516 . . . . . 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 5895 . . . . . . 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 7839 . . . . . . . . . . 11 1o = {∅}
268fvexi 6447 . . . . . . . . . . 11 𝐵 ∈ V
27 0ex 5014 . . . . . . . . . . 11 ∅ ∈ V
28 eqid 2825 . . . . . . . . . . 11 (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))
2925, 26, 27, 28mapsncnv 8171 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅)) = (𝑧𝐵 ↦ (1o × {𝑧}))
3029coeq1i 5514 . . . . . . . . 9 ((𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) = ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅)))
3125, 26, 27, 28mapsnf1o2 8172 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅)):(𝐵𝑚 1o)–1-1-onto𝐵
32 f1ococnv1 6406 . . . . . . . . . 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, 33syl5eqr 2875 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵𝑚 1o)))
3534coeq2d 5517 . . . . . . 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 2873 . . . . . 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 2825 . . . . . . . 8 (𝑅s (𝐵𝑚 1o)) = (𝑅s (𝐵𝑚 1o))
38 eqid 2825 . . . . . . . 8 (Base‘(𝑅s (𝐵𝑚 1o))) = (Base‘(𝑅s (𝐵𝑚 1o)))
39 simpl 476 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → 𝑅 ∈ CRing)
40 ovexd 6939 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (𝐵𝑚 1o) ∈ V)
41 1on 7833 . . . . . . . . . . 11 1o ∈ On
4218, 8, 19, 37evlrhm 19885 . . . . . . . . . . 11 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1o))))
4341, 42mpan 681 . . . . . . . . . 10 (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅s (𝐵𝑚 1o))))
4421, 38rhmf 19082 . . . . . . . . . 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 6608 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ (Base‘(𝑅s (𝐵𝑚 1o))))
4737, 8, 38, 39, 40, 46pwselbas 16502 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵𝑚 1o)⟶𝐵)
48 fcoi1 6315 . . . . . . 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 2865 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
5145ffnd 6279 . . . . . . 7 (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1𝑅)))
52 fnfvelrn 6605 . . . . . . 7 (((1o eval 𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
5351, 52sylan 575 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
54 mpfpf1.q . . . . . 6 𝐸 = ran (1o eval 𝑅)
5553, 54syl6eleqr 2917 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
5650, 55eqeltrd 2906 . . . 4 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
57 coeq1 5512 . . . . 5 (((eval1𝑅)‘𝑦) = 𝐹 → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))))
5857eleq1d 2891 . . . 4 (((eval1𝑅)‘𝑦) = 𝐹 → ((((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
5956, 58syl5ibcom 237 . . 3 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
6059rexlimdva 3240 . 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 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∃wrex 3118  Vcvv 3414  ∅c0 4144  {csn 4397   ↦ cmpt 4952   I cid 5249   × cxp 5340  ◡ccnv 5341  ran crn 5343   ↾ cres 5344   ∘ ccom 5346  Oncon0 5963   Fn wfn 6118  ⟶wf 6119  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905  1oc1o 7819   ↑𝑚 cmap 8122  Basecbs 16222   ↑s cpws 16460  CRingccrg 18902   RingHom crh 19068   mPoly cmpl 19714   eval cevl 19865  PwSer1cps1 19905  Poly1cpl1 19907  eval1ce1 20039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-ofr 7158  df-om 7327  df-1st 7428  df-2nd 7429  df-supp 7560  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fsupp 8545  df-sup 8617  df-oi 8684  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-fzo 12761  df-seq 13096  df-hash 13411  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-hom 16329  df-cco 16330  df-0g 16455  df-gsum 16456  df-prds 16461  df-pws 16463  df-mre 16599  df-mrc 16600  df-acs 16602  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-submnd 17689  df-grp 17779  df-minusg 17780  df-sbg 17781  df-mulg 17895  df-subg 17942  df-ghm 18009  df-cntz 18100  df-cmn 18548  df-abl 18549  df-mgp 18844  df-ur 18856  df-srg 18860  df-ring 18903  df-cring 18904  df-rnghom 19071  df-subrg 19134  df-lmod 19221  df-lss 19289  df-lsp 19331  df-assa 19673  df-asp 19674  df-ascl 19675  df-psr 19717  df-mvr 19718  df-mpl 19719  df-opsr 19721  df-evls 19866  df-evl 19867  df-psr1 19910  df-ply1 19912  df-evl1 20041 This theorem is referenced by:  pf1ind  20079
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