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Theorem pf1mpf 20507
 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 20504 . 2 (𝐹𝑄𝑅 ∈ CRing)
3 id 22 . . . 4 (𝐹𝑄𝐹𝑄)
43, 1eleqtrdi 2921 . . 3 (𝐹𝑄𝐹 ∈ ran (eval1𝑅))
5 eqid 2819 . . . . . 6 (eval1𝑅) = (eval1𝑅)
6 eqid 2819 . . . . . 6 (Poly1𝑅) = (Poly1𝑅)
7 eqid 2819 . . . . . 6 (𝑅s 𝐵) = (𝑅s 𝐵)
8 pf1f.b . . . . . 6 𝐵 = (Base‘𝑅)
95, 6, 7, 8evl1rhm 20487 . . . . 5 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
102, 9syl 17 . . . 4 (𝐹𝑄 → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
11 eqid 2819 . . . . 5 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2819 . . . . 5 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
1311, 12rhmf 19470 . . . 4 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
14 ffn 6507 . . . 4 ((eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)) → (eval1𝑅) Fn (Base‘(Poly1𝑅)))
15 fvelrnb 6719 . . . 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 234 . 2 (𝐹𝑄 → ∃𝑦 ∈ (Base‘(Poly1𝑅))((eval1𝑅)‘𝑦) = 𝐹)
18 eqid 2819 . . . . . . . 8 (1o eval 𝑅) = (1o eval 𝑅)
19 eqid 2819 . . . . . . . 8 (1o mPoly 𝑅) = (1o mPoly 𝑅)
20 eqid 2819 . . . . . . . . 9 (PwSer1𝑅) = (PwSer1𝑅)
216, 20, 11ply1bas 20355 . . . . . . . 8 (Base‘(Poly1𝑅)) = (Base‘(1o mPoly 𝑅))
225, 18, 8, 19, 21evl1val 20484 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((eval1𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧𝐵 ↦ (1o × {𝑧}))))
2322coeq1d 5725 . . . . . 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 6111 . . . . . . 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 8108 . . . . . . . . . . 11 1o = {∅}
268fvexi 6677 . . . . . . . . . . 11 𝐵 ∈ V
27 0ex 5202 . . . . . . . . . . 11 ∅ ∈ V
28 eqid 2819 . . . . . . . . . . 11 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))
2925, 26, 27, 28mapsncnv 8449 . . . . . . . . . 10 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) = (𝑧𝐵 ↦ (1o × {𝑧}))
3029coeq1i 5723 . . . . . . . . 9 ((𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)))
3125, 26, 27, 28mapsnf1o2 8450 . . . . . . . . . 10 (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅)):(𝐵m 1o)–1-1-onto𝐵
32 f1ococnv1 6636 . . . . . . . . . 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 2868 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((𝑧𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵m 1o)))
3534coeq2d 5726 . . . . . . 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 2866 . . . . . 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 2819 . . . . . . . 8 (𝑅s (𝐵m 1o)) = (𝑅s (𝐵m 1o))
38 eqid 2819 . . . . . . . 8 (Base‘(𝑅s (𝐵m 1o))) = (Base‘(𝑅s (𝐵m 1o)))
39 simpl 485 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → 𝑅 ∈ CRing)
40 ovexd 7183 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (𝐵m 1o) ∈ V)
41 1on 8101 . . . . . . . . . . 11 1o ∈ On
4218, 8, 19, 37evlrhm 20301 . . . . . . . . . . 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 19470 . . . . . . . . . 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 6844 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ (Base‘(𝑅s (𝐵m 1o))))
4737, 8, 38, 39, 40, 46pwselbas 16754 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵m 1o)⟶𝐵)
48 fcoi1 6545 . . . . . . 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 2858 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
5145ffnd 6508 . . . . . . 7 (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1𝑅)))
52 fnfvelrn 6841 . . . . . . 7 (((1o eval 𝑅) Fn (Base‘(Poly1𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
5351, 52sylan 582 . . . . . 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 2922 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
5650, 55eqeltrd 2911 . . . 4 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
57 coeq1 5721 . . . . 5 (((eval1𝑅)‘𝑦) = 𝐹 → (((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))))
5857eleq1d 2895 . . . 4 (((eval1𝑅)‘𝑦) = 𝐹 → ((((eval1𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
5956, 58syl5ibcom 247 . . 3 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1𝑅))) → (((eval1𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
6059rexlimdva 3282 . 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 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∃wrex 3137  Vcvv 3493  ∅c0 4289  {csn 4559   ↦ cmpt 5137   I cid 5452   × cxp 5546  ◡ccnv 5547  ran crn 5549   ↾ cres 5550   ∘ ccom 5552  Oncon0 6184   Fn wfn 6343  ⟶wf 6344  –1-1-onto→wf1o 6347  ‘cfv 6348  (class class class)co 7148  1oc1o 8087   ↑m cmap 8398  Basecbs 16475   ↑s cpws 16712  CRingccrg 19290   RingHom crh 19456   mPoly cmpl 20125   eval cevl 20277  PwSer1cps1 20335  Poly1cpl1 20337  eval1ce1 20469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-ofr 7402  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-srg 19248  df-ring 19291  df-cring 19292  df-rnghom 19459  df-subrg 19525  df-lmod 19628  df-lss 19696  df-lsp 19736  df-assa 20077  df-asp 20078  df-ascl 20079  df-psr 20128  df-mvr 20129  df-mpl 20130  df-opsr 20132  df-evls 20278  df-evl 20279  df-psr1 20340  df-ply1 20342  df-evl1 20471 This theorem is referenced by:  pf1ind  20510
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