Theorem pf1mpf 19716

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 19713	. 2 ⊢ (𝐹 ∈ 𝑄 → 𝑅 ∈ CRing)
3		id 22	. . . 4 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ 𝑄)
4	3, 1	syl6eleq 2711	. . 3 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ ran (eval1‘𝑅))
5		eqid 2622	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
6		eqid 2622	. . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
7		eqid 2622	. . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
8		pf1f.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	5, 6, 7, 8	evl1rhm 19696	. . . . 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 2622	. . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2622	. . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
13	11, 12	rhmf 18726	. . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
14		ffn 6045	. . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)))
15		fvelrnb 6243	. . . 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 222	. 2 ⊢ (𝐹 ∈ 𝑄 → ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹)
18		eqid 2622	. . . . . . . 8 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
19		eqid 2622	. . . . . . . 8 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
20		eqid 2622	. . . . . . . . 9 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
21	6, 20, 11	ply1bas 19565	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
22	5, 18, 8, 19, 21	evl1val 19693	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑦) = (((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧}))))
23	22	coeq1d 5283	. . . . . 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‘∅))))
24		coass 5654	. . . . . . 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‘∅))))
25		df1o2 7572	. . . . . . . . . . 11 ⊢ 1𝑜 = {∅}
26		fvex 6201	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
27	8, 26	eqeltri 2697	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
28		0ex 4790	. . . . . . . . . . 11 ⊢ ∅ ∈ V
29		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))
30	25, 27, 28, 29	mapsncnv 7904	. . . . . . . . . 10 ⊢ ◡(𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧}))
31	30	coeq1i 5281	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ((𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)))
32	25, 27, 28, 29	mapsnf1o2 7905	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)):(𝐵 ↑𝑚 1𝑜)–1-1-onto→𝐵
33		f1ococnv1 6165	. . . . . . . . . 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)))
34	32, 33	mp1i 13	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (◡(𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑𝑚 1𝑜)))
35	31, 34	syl5eqr 2670	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑𝑚 1𝑜)))
36	35	coeq2d 5284	. . . . . . 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))))
37	24, 36	syl5eq 2668	. . . . . 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))))
38		eqid 2622	. . . . . . . 8 ⊢ (𝑅 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑅 ↑s (𝐵 ↑𝑚 1𝑜))
39		eqid 2622	. . . . . . . 8 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))) = (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜)))
40		simpl 473	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → 𝑅 ∈ CRing)
41		ovexd 6680	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (𝐵 ↑𝑚 1𝑜) ∈ V)
42		1on 7567	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
43	18, 8, 19, 38	evlrhm 19525	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
44	42, 43	mpan 706	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
45	21, 39	rhmf 18726	. . . . . . . . . 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))))
46	44, 45	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (1𝑜 eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
47	46	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
48	38, 8, 39, 40, 41, 47	pwselbas 16149	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦):(𝐵 ↑𝑚 1𝑜)⟶𝐵)
49		fcoi1 6078	. . . . . . 7 ⊢ (((1𝑜 eval 𝑅)‘𝑦):(𝐵 ↑𝑚 1𝑜)⟶𝐵 → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
50	48, 49	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
51	23, 37, 50	3eqtrd 2660	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ((1𝑜 eval 𝑅)‘𝑦))
52		ffn 6045	. . . . . . . 8 ⊢ ((1𝑜 eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))) → (1𝑜 eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
53	46, 52	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (1𝑜 eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
54		fnfvelrn 6356	. . . . . . 7 ⊢ (((1𝑜 eval 𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
55	53, 54	sylan 488	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
56		mpfpf1.q	. . . . . 6 ⊢ 𝐸 = ran (1𝑜 eval 𝑅)
57	55, 56	syl6eleqr 2712	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ 𝐸)
58	51, 57	eqeltrd 2701	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
59		coeq1 5279	. . . . 5 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))))
60	59	eleq1d 2686	. . . 4 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → ((((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
61	58, 60	syl5ibcom 235	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
62	61	rexlimdva 3031	. 2 ⊢ (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
63	2, 17, 62	sylc 65	1 ⊢ (𝐹 ∈ 𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200  ∅c0 3915  {csn 4177   ↦ cmpt 4729   I cid 5023   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   ∘ ccom 5118  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857  Basecbs 15857   ↑_s cpws 16107  CRingccrg 18548   RingHom crh 18712   mPoly cmpl 19353   eval cevl 19505  PwSer₁cps1 19545  Poly₁cpl1 19547  eval₁ce1 19679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-ply1 19552  df-evl1 19681

This theorem is referenced by:  pf1ind  19719