Theorem pf1mpf 22091

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 22088	. 2 ⊢ (𝐹 ∈ 𝑄 → 𝑅 ∈ CRing)
3		id 22	. . . 4 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ 𝑄)
4	3, 1	eleqtrdi 2841	. . 3 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ ran (eval1‘𝑅))
5		eqid 2730	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
6		eqid 2730	. . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
7		eqid 2730	. . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
8		pf1f.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	5, 6, 7, 8	evl1rhm 22071	. . . . 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 2730	. . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2730	. . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
13	11, 12	rhmf 20376	. . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
14		ffn 6716	. . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)))
15		fvelrnb 6951	. . . 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 231	. 2 ⊢ (𝐹 ∈ 𝑄 → ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹)
18		eqid 2730	. . . . . . . 8 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
19		eqid 2730	. . . . . . . 8 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
20		eqid 2730	. . . . . . . . 9 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
21	6, 20, 11	ply1bas 21938	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅))
22	5, 18, 8, 19, 21	evl1val 22068	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))))
23	22	coeq1d 5860	. . . . . 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 6263	. . . . . . 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 8475	. . . . . . . . . . 11 ⊢ 1o = {∅}
26	8	fvexi 6904	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
27		0ex 5306	. . . . . . . . . . 11 ⊢ ∅ ∈ V
28		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))
29	25, 26, 27, 28	mapsncnv 8889	. . . . . . . . . 10 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))
30	29	coeq1i 5858	. . . . . . . . 9 ⊢ (◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)))
31	25, 26, 27, 28	mapsnf1o2 8890	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
32		f1ococnv1 6861	. . . . . . . . . 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)))
33	31, 32	mp1i 13	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
34	30, 33	eqtr3id 2784	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
35	34	coeq2d 5861	. . . . . . 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))))
36	24, 35	eqtrid 2782	. . . . . 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))))
37		eqid 2730	. . . . . . . 8 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o))
38		eqid 2730	. . . . . . . 8 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))
39		simpl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → 𝑅 ∈ CRing)
40		ovexd 7446	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (𝐵 ↑m 1o) ∈ V)
41		1on 8480	. . . . . . . . . . 11 ⊢ 1o ∈ On
42	18, 8, 19, 37	evlrhm 21878	. . . . . . . . . . 11 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
43	41, 42	mpan 686	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
44	21, 38	rhmf 20376	. . . . . . . . . 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))))
45	43, 44	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s (𝐵 ↑m 1o))))
46	45	ffvelcdmda 7085	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))))
47	37, 8, 38, 39, 40, 46	pwselbas 17439	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵 ↑m 1o)⟶𝐵)
48		fcoi1 6764	. . . . . . 7 ⊢ (((1o eval 𝑅)‘𝑦):(𝐵 ↑m 1o)⟶𝐵 → (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑m 1o))) = ((1o eval 𝑅)‘𝑦))
49	47, 48	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((1o eval 𝑅)‘𝑦) ∘ ( I ↾ (𝐵 ↑m 1o))) = ((1o eval 𝑅)‘𝑦))
50	23, 36, 49	3eqtrd 2774	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
51	45	ffnd 6717	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
52		fnfvelrn 7081	. . . . . . 7 ⊢ (((1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
53	51, 52	sylan 578	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
54		mpfpf1.q	. . . . . 6 ⊢ 𝐸 = ran (1o eval 𝑅)
55	53, 54	eleqtrrdi 2842	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
56	50, 55	eqeltrd 2831	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
57		coeq1 5856	. . . . 5 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))))
58	57	eleq1d 2816	. . . 4 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → ((((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸 ↔ (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
59	56, 58	syl5ibcom 244	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
60	59	rexlimdva 3153	. 2 ⊢ (𝑅 ∈ CRing → (∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
61	2, 17, 60	sylc 65	1 ⊢ (𝐹 ∈ 𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∃wrex 3068  Vcvv 3472  ∅c0 4321  {csn 4627   ↦ cmpt 5230   I cid 5572   × cxp 5673  ^◡ccnv 5674  ran crn 5676   ↾ cres 5677   ∘ ccom 5679  Oncon0 6363   Fn wfn 6537  ⟶wf 6538  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7411  1_oc1o 8461   ↑_m cmap 8822  Basecbs 17148   ↑_s cpws 17396  CRingccrg 20128   RingHom crh 20360   mPoly cmpl 21678   eval cevl 21853  PwSer₁cps1 21918  Poly₁cpl1 21920  eval₁ce1 22053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-ofr 7673  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-hom 17225  df-cco 17226  df-0g 17391  df-gsum 17392  df-prds 17397  df-pws 17399  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-ghm 19128  df-cntz 19222  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-srg 20081  df-ring 20129  df-cring 20130  df-rhm 20363  df-subrng 20434  df-subrg 20459  df-lmod 20616  df-lss 20687  df-lsp 20727  df-assa 21627  df-asp 21628  df-ascl 21629  df-psr 21681  df-mvr 21682  df-mpl 21683  df-opsr 21685  df-evls 21854  df-evl 21855  df-psr1 21923  df-ply1 21925  df-evl1 22055

This theorem is referenced by:  pf1ind  22094