Theorem pf1mpf 19537

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 19534	. 2 ⊢ (𝐹 ∈ 𝑄 → 𝑅 ∈ CRing)
3		id 22	. . . 4 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ 𝑄)
4	3, 1	syl6eleq 2698	. . 3 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ ran (eval1‘𝑅))
5		eqid 2610	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
6		eqid 2610	. . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
7		eqid 2610	. . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
8		pf1f.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	5, 6, 7, 8	evl1rhm 19517	. . . . 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 2610	. . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2610	. . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
13	11, 12	rhmf 18549	. . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
14		ffn 5958	. . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)))
15		fvelrnb 6153	. . . 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 221	. 2 ⊢ (𝐹 ∈ 𝑄 → ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹)
18		eqid 2610	. . . . . . . 8 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
19		eqid 2610	. . . . . . . 8 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
20		eqid 2610	. . . . . . . . 9 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
21	6, 20, 11	ply1bas 19386	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
22	5, 18, 8, 19, 21	evl1val 19514	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑦) = (((1𝑜 eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧}))))
23	22	coeq1d 5205	. . . . . 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 5571	. . . . . . 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 7459	. . . . . . . . . . 11 ⊢ 1𝑜 = {∅}
26		fvex 6113	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) ∈ V
27	8, 26	eqeltri 2684	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
28		0ex 4718	. . . . . . . . . . 11 ⊢ ∅ ∈ V
29		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))
30	25, 27, 28, 29	mapsncnv 7790	. . . . . . . . . 10 ⊢ ◡(𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)) = (𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧}))
31	30	coeq1i 5203	. . . . . . . . 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 7791	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅)):(𝐵 ↑𝑚 1𝑜)–1-1-onto→𝐵
33		f1ococnv1 6078	. . . . . . . . . 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 2658	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((𝑧 ∈ 𝐵 ↦ (1𝑜 × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑𝑚 1𝑜)))
36	35	coeq2d 5206	. . . . . . 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 2656	. . . . . 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 2610	. . . . . . . 8 ⊢ (𝑅 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑅 ↑s (𝐵 ↑𝑚 1𝑜))
39		eqid 2610	. . . . . . . 8 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))) = (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜)))
40		simpl 472	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → 𝑅 ∈ CRing)
41		ovex 6577	. . . . . . . . 9 ⊢ (𝐵 ↑𝑚 1𝑜) ∈ V
42	41	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (𝐵 ↑𝑚 1𝑜) ∈ V)
43		1on 7454	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
44	18, 8, 19, 38	evlrhm 19346	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
45	43, 44	mpan 702	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (1𝑜 eval 𝑅) ∈ ((1𝑜 mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
46	21, 39	rhmf 18549	. . . . . . . . . 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))))
47	45, 46	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (1𝑜 eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
48	47	ffvelrnda 6267	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ (Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))))
49	38, 8, 39, 40, 42, 48	pwselbas 15972	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦):(𝐵 ↑𝑚 1𝑜)⟶𝐵)
50		fcoi1 5991	. . . . . . 7 ⊢ (((1𝑜 eval 𝑅)‘𝑦):(𝐵 ↑𝑚 1𝑜)⟶𝐵 → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
51	49, 50	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((1𝑜 eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑𝑚 1𝑜))) = ((1𝑜 eval 𝑅)‘𝑦))
52	23, 37, 51	3eqtrd 2648	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = ((1𝑜 eval 𝑅)‘𝑦))
53		ffn 5958	. . . . . . . 8 ⊢ ((1𝑜 eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑𝑚 1𝑜))) → (1𝑜 eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
54	47, 53	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (1𝑜 eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
55		fnfvelrn 6264	. . . . . . 7 ⊢ (((1𝑜 eval 𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
56	54, 55	sylan 487	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ ran (1𝑜 eval 𝑅))
57		mpfpf1.q	. . . . . 6 ⊢ 𝐸 = ran (1𝑜 eval 𝑅)
58	56, 57	syl6eleqr 2699	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1𝑜 eval 𝑅)‘𝑦) ∈ 𝐸)
59	52, 58	eqeltrd 2688	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)
60		coeq1 5201	. . . . 5 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))))
61	60	eleq1d 2672	. . . 4 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → ((((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
62	59, 61	syl5ibcom 234	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
63	62	rexlimdva 3013	. 2 ⊢ (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸))
64	2, 17, 63	sylc 63	1 ⊢ (𝐹 ∈ 𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑𝑚 1𝑜) ↦ (𝑥‘∅))) ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125   ↦ cmpt 4643   I cid 4948   × cxp 5036  ^◡ccnv 5037  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  Basecbs 15695   ↑_s cpws 15930  CRingccrg 18371   RingHom crh 18535   mPoly cmpl 19174   eval cevl 19326  PwSer₁cps1 19366  Poly₁cpl1 19368  eval₁ce1 19500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-rnghom 18538  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-assa 19133  df-asp 19134  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-evls 19327  df-evl 19328  df-psr1 19371  df-ply1 19373  df-evl1 19502

This theorem is referenced by:  pf1ind  19540