Theorem pf1mpf 22268

Description: Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pf1rcl.q	⊢ 𝑄 = ran (eval1‘𝑅)
pf1f.b	⊢ 𝐵 = (Base‘𝑅)
mpfpf1.q	⊢ 𝐸 = ran (1o eval 𝑅)

Assertion

Ref	Expression
pf1mpf	⊢ (𝐹 ∈ 𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 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.

Step	Hyp	Ref	Expression
1		pf1rcl.q	. . 3 ⊢ 𝑄 = ran (eval1‘𝑅)
2	1	pf1rcl 22265	. 2 ⊢ (𝐹 ∈ 𝑄 → 𝑅 ∈ CRing)
3		id 22	. . . 4 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ 𝑄)
4	3, 1	eleqtrdi 2843	. . 3 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ ran (eval1‘𝑅))
5		eqid 2733	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
6		eqid 2733	. . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
7		eqid 2733	. . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
8		pf1f.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	5, 6, 7, 8	evl1rhm 22248	. . . . 5 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
10	2, 9	syl 17	. . . 4 ⊢ (𝐹 ∈ 𝑄 → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)))
11		eqid 2733	. . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2733	. . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
13	11, 12	rhmf 20404	. . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
14		ffn 6656	. . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)))
15		fvelrnb 6888	. . . 4 ⊢ ((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) → (𝐹 ∈ ran (eval1‘𝑅) ↔ ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹))
16	10, 13, 14, 15	4syl 19	. . 3 ⊢ (𝐹 ∈ 𝑄 → (𝐹 ∈ ran (eval1‘𝑅) ↔ ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹))
17	4, 16	mpbid 232	. 2 ⊢ (𝐹 ∈ 𝑄 → ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹)
18		eqid 2733	. . . . . . . 8 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
19		eqid 2733	. . . . . . . 8 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
20	6, 11	ply1bas 22108	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅))
21	5, 18, 8, 19, 20	evl1val 22245	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))))
22	21	coeq1d 5805	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))))
23		coass 6218	. . . . . . 7 ⊢ ((((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = (((1o eval 𝑅)‘𝑦) ∘ ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))))
24		df1o2 8398	. . . . . . . . . . 11 ⊢ 1o = {∅}
25	8	fvexi 6842	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
26		0ex 5247	. . . . . . . . . . 11 ⊢ ∅ ∈ V
27		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))
28	24, 25, 26, 27	mapsncnv 8823	. . . . . . . . . 10 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))
29	28	coeq1i 5803	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)))
30	24, 25, 26, 27	mapsnf1o2 8824	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
31		f1ococnv1 6797	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵 → (◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
32	30, 31	mp1i 13	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
33	29, 32	eqtr3id 2782	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
34	33	coeq2d 5806	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑦) ∘ ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)))) = (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑m 1o))))
35	23, 34	eqtrid 2780	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑m 1o))))
36		eqid 2733	. . . . . . . 8 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o))
37		eqid 2733	. . . . . . . 8 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))
38		simpl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → 𝑅 ∈ CRing)
39		ovexd 7387	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (𝐵 ↑m 1o) ∈ V)
40		1on 8403	. . . . . . . . . . 11 ⊢ 1o ∈ On
41	18, 8, 19, 36	evlrhm 22032	. . . . . . . . . . 11 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
42	40, 41	mpan 690	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
43	20, 37	rhmf 20404	. . . . . . . . . 10 ⊢ ((1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))) → (1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))))
44	42, 43	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))))
45	44	ffvelcdmda 7023	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))))
46	36, 8, 37, 38, 39, 45	pwselbas 17395	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵 ↑m 1o)⟶𝐵)
47		fcoi1 6702	. . . . . . 7 ⊢ (((1o eval 𝑅)‘𝑦):(𝐵 ↑m 1o)⟶𝐵 → (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑m 1o))) = ((1o eval 𝑅)‘𝑦))
48	46, 47	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑m 1o))) = ((1o eval 𝑅)‘𝑦))
49	22, 35, 48	3eqtrd 2772	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
50	44	ffnd 6657	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
51		fnfvelrn 7019	. . . . . . 7 ⊢ (((1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
52	50, 51	sylan 580	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
53		mpfpf1.q	. . . . . 6 ⊢ 𝐸 = ran (1o eval 𝑅)
54	52, 53	eleqtrrdi 2844	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
55	49, 54	eqeltrd 2833	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
56		coeq1 5801	. . . . 5 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))))
57	56	eleq1d 2818	. . . 4 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → ((((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
58	55, 57	syl5ibcom 245	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
59	58	rexlimdva 3134	. 2 ⊢ (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
60	2, 17, 59	sylc 65	1 ⊢ (𝐹 ∈ 𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437  ∅c0 4282  {csn 4575   ↦ cmpt 5174   I cid 5513   × cxp 5617  ^◡ccnv 5618  ran crn 5620   ↾ cres 5621   ∘ ccom 5623  Oncon0 6311   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7352  1_oc1o 8384   ↑_m cmap 8756  Basecbs 17122   ↑_s cpws 17352  CRingccrg 20154   RingHom crh 20389   mPoly cmpl 21845   eval cevl 22009  Poly₁cpl1 22090  eval₁ce1 22230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-ofr 7617  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-pm 8759  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-sup 9333  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-hom 17187  df-cco 17188  df-0g 17347  df-gsum 17348  df-prds 17353  df-pws 17355  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-submnd 18694  df-grp 18851  df-minusg 18852  df-sbg 18853  df-mulg 18983  df-subg 19038  df-ghm 19127  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-srg 20107  df-ring 20155  df-cring 20156  df-rhm 20392  df-subrng 20463  df-subrg 20487  df-lmod 20797  df-lss 20867  df-lsp 20907  df-assa 21792  df-asp 21793  df-ascl 21794  df-psr 21848  df-mvr 21849  df-mpl 21850  df-opsr 21852  df-evls 22010  df-evl 22011  df-psr1 22093  df-ply1 22095  df-evl1 22232

This theorem is referenced by:  pf1ind  22271