Theorem pf1mpf 20511

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 20508	. 2 ⊢ (𝐹 ∈ 𝑄 → 𝑅 ∈ CRing)
3		id 22	. . . 4 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ 𝑄)
4	3, 1	eleqtrdi 2922	. . 3 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ ran (eval1‘𝑅))
5		eqid 2820	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
6		eqid 2820	. . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
7		eqid 2820	. . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
8		pf1f.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	5, 6, 7, 8	evl1rhm 20491	. . . . 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 2820	. . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2820	. . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
13	11, 12	rhmf 19474	. . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
14		ffn 6511	. . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)))
15		fvelrnb 6723	. . . 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 234	. 2 ⊢ (𝐹 ∈ 𝑄 → ∃𝑦 ∈ (Base‘(Poly1‘𝑅))((eval1‘𝑅)‘𝑦) = 𝐹)
18		eqid 2820	. . . . . . . 8 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
19		eqid 2820	. . . . . . . 8 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
20		eqid 2820	. . . . . . . . 9 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
21	6, 20, 11	ply1bas 20359	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅))
22	5, 18, 8, 19, 21	evl1val 20488	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))))
23	22	coeq1d 5729	. . . . . 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 6115	. . . . . . 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 8113	. . . . . . . . . . 11 ⊢ 1o = {∅}
26	8	fvexi 6681	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
27		0ex 5208	. . . . . . . . . . 11 ⊢ ∅ ∈ V
28		eqid 2820	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))
29	25, 26, 27, 28	mapsncnv 8454	. . . . . . . . . 10 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))
30	29	coeq1i 5727	. . . . . . . . 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 8455	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
32		f1ococnv1 6640	. . . . . . . . . 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	syl5eqr 2869	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
35	34	coeq2d 5730	. . . . . . 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	syl5eq 2867	. . . . . 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 2820	. . . . . . . 8 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o))
38		eqid 2820	. . . . . . . 8 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))
39		simpl 485	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → 𝑅 ∈ CRing)
40		ovexd 7188	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (𝐵 ↑m 1o) ∈ V)
41		1on 8106	. . . . . . . . . . 11 ⊢ 1o ∈ On
42	18, 8, 19, 37	evlrhm 20305	. . . . . . . . . . 11 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
43	41, 42	mpan 688	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
44	21, 38	rhmf 19474	. . . . . . . . . 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	ffvelrnda 6848	. . . . . . . 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 16758	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵 ↑m 1o)⟶𝐵)
48		fcoi1 6549	. . . . . . 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 2859	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
51	45	ffnd 6512	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
52		fnfvelrn 6845	. . . . . . 7 ⊢ (((1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
53	51, 52	sylan 582	. . . . . 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 2923	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
56	50, 55	eqeltrd 2912	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
57		coeq1 5725	. . . . 5 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))))
58	57	eleq1d 2896	. . . 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 247	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) = 𝐹 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸))
60	59	rexlimdva 3283	. 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 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∃wrex 3138  Vcvv 3493  ∅c0 4288  {csn 4564   ↦ cmpt 5143   I cid 5456   × cxp 5550  ^◡ccnv 5551  ran crn 5553   ↾ cres 5554   ∘ ccom 5556  Oncon0 6188   Fn wfn 6347  ⟶wf 6348  –1-1-onto→wf1o 6351  ‘cfv 6352  (class class class)co 7153  1_oc1o 8092   ↑_m cmap 8403  Basecbs 16479   ↑_s cpws 16716  CRingccrg 19294   RingHom crh 19460   mPoly cmpl 20129   eval cevl 20281  PwSer₁cps1 20339  Poly₁cpl1 20341  eval₁ce1 20473

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458  ax-cnex 10590  ax-resscn 10591  ax-1cn 10592  ax-icn 10593  ax-addcl 10594  ax-addrcl 10595  ax-mulcl 10596  ax-mulrcl 10597  ax-mulcom 10598  ax-addass 10599  ax-mulass 10600  ax-distr 10601  ax-i2m1 10602  ax-1ne0 10603  ax-1rid 10604  ax-rnegex 10605  ax-rrecex 10606  ax-cnre 10607  ax-pre-lttri 10608  ax-pre-lttrn 10609  ax-pre-ltadd 10610  ax-pre-mulgt0 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4465  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4836  df-int 4874  df-iun 4918  df-iin 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5457  df-eprel 5462  df-po 5471  df-so 5472  df-fr 5511  df-se 5512  df-we 5513  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7111  df-ov 7156  df-oprab 7157  df-mpo 7158  df-of 7406  df-ofr 7407  df-om 7578  df-1st 7686  df-2nd 7687  df-supp 7828  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-2o 8100  df-oadd 8103  df-er 8286  df-map 8405  df-pm 8406  df-ixp 8459  df-en 8507  df-dom 8508  df-sdom 8509  df-fin 8510  df-fsupp 8831  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10674  df-mnf 10675  df-xr 10676  df-ltxr 10677  df-le 10678  df-sub 10869  df-neg 10870  df-nn 11636  df-2 11698  df-3 11699  df-4 11700  df-5 11701  df-6 11702  df-7 11703  df-8 11704  df-9 11705  df-n0 11896  df-z 11980  df-dec 12097  df-uz 12242  df-fz 12891  df-fzo 13032  df-seq 13368  df-hash 13689  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-hom 16585  df-cco 16586  df-0g 16711  df-gsum 16712  df-prds 16717  df-pws 16719  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-submnd 17953  df-grp 18102  df-minusg 18103  df-sbg 18104  df-mulg 18221  df-subg 18272  df-ghm 18352  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19236  df-ur 19248  df-srg 19252  df-ring 19295  df-cring 19296  df-rnghom 19463  df-subrg 19529  df-lmod 19632  df-lss 19700  df-lsp 19740  df-assa 20081  df-asp 20082  df-ascl 20083  df-psr 20132  df-mvr 20133  df-mpl 20134  df-opsr 20136  df-evls 20282  df-evl 20283  df-psr1 20344  df-ply1 20346  df-evl1 20475

This theorem is referenced by:  pf1ind  20514