Theorem pf1mpf 22300

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 22297	. 2 ⊢ (𝐹 ∈ 𝑄 → 𝑅 ∈ CRing)
3		id 22	. . . 4 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ 𝑄)
4	3, 1	eleqtrdi 2847	. . 3 ⊢ (𝐹 ∈ 𝑄 → 𝐹 ∈ ran (eval1‘𝑅))
5		eqid 2737	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
6		eqid 2737	. . . . . 6 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
7		eqid 2737	. . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵)
8		pf1f.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
9	5, 6, 7, 8	evl1rhm 22280	. . . . 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 2737	. . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
12		eqid 2737	. . . . 5 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵))
13	11, 12	rhmf 20424	. . . 4 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)))
14		ffn 6663	. . . 4 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)))
15		fvelrnb 6895	. . . 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 2737	. . . . . . . 8 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
19		eqid 2737	. . . . . . . 8 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
20	6, 11	ply1bas 22139	. . . . . . . 8 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅))
21	5, 18, 8, 19, 20	evl1val 22277	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘𝑦) = (((1o eval 𝑅)‘𝑦) ∘ (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))))
22	21	coeq1d 5811	. . . . . 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 6225	. . . . . . 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 8406	. . . . . . . . . . 11 ⊢ 1o = {∅}
25	8	fvexi 6849	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
26		0ex 5253	. . . . . . . . . . 11 ⊢ ∅ ∈ V
27		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))
28	24, 25, 26, 27	mapsncnv 8835	. . . . . . . . . 10 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)) = (𝑧 ∈ 𝐵 ↦ (1o × {𝑧}))
29	28	coeq1i 5809	. . . . . . . . 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 8836	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
31		f1ococnv1 6804	. . . . . . . . . 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 2786	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((𝑧 ∈ 𝐵 ↦ (1o × {𝑧})) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ( I ↾ (𝐵 ↑m 1o)))
34	33	coeq2d 5812	. . . . . . 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 2784	. . . . . 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 2737	. . . . . . . 8 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o))
37		eqid 2737	. . . . . . . 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 7395	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (𝐵 ↑m 1o) ∈ V)
40		1on 8411	. . . . . . . . . . 11 ⊢ 1o ∈ On
41	18, 8, 19, 36	evlrhm 22060	. . . . . . . . . . 11 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
42	40, 41	mpan 691	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o))))
43	20, 37	rhmf 20424	. . . . . . . . . 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 7031	. . . . . . . 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 17413	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦):(𝐵 ↑m 1o)⟶𝐵)
47		fcoi1 6709	. . . . . . 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 2776	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = ((1o eval 𝑅)‘𝑦))
50	44	ffnd 6664	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)))
51		fnfvelrn 7027	. . . . . . 7 ⊢ (((1o eval 𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ ran (1o eval 𝑅))
52	50, 51	sylan 581	. . . . . 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 2848	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → ((1o eval 𝑅)‘𝑦) ∈ 𝐸)
55	49, 54	eqeltrd 2837	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘(Poly1‘𝑅))) → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
56		coeq1 5807	. . . . 5 ⊢ (((eval1‘𝑅)‘𝑦) = 𝐹 → (((eval1‘𝑅)‘𝑦) ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) = (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))))
57	56	eleq1d 2822	. . . 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 3138	. 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 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3441  ∅c0 4286  {csn 4581   ↦ cmpt 5180   I cid 5519   × cxp 5623  ^◡ccnv 5624  ran crn 5626   ↾ cres 5627   ∘ ccom 5629  Oncon0 6318   Fn wfn 6488  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  1_oc1o 8392   ↑_m cmap 8767  Basecbs 17140   ↑_s cpws 17370  CRingccrg 20173   RingHom crh 20409   mPoly cmpl 21866   eval cevl 22032  Poly₁cpl1 22121  eval₁ce1 22262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-dec 12612  df-uz 12756  df-fz 13428  df-fzo 13575  df-seq 13929  df-hash 14258  df-struct 17078  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-hom 17205  df-cco 17206  df-0g 17365  df-gsum 17366  df-prds 17371  df-pws 17373  df-mre 17509  df-mrc 17510  df-acs 17512  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-mhm 18712  df-submnd 18713  df-grp 18870  df-minusg 18871  df-sbg 18872  df-mulg 19002  df-subg 19057  df-ghm 19146  df-cntz 19250  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-srg 20126  df-ring 20174  df-cring 20175  df-rhm 20412  df-subrng 20483  df-subrg 20507  df-lmod 20817  df-lss 20887  df-lsp 20927  df-assa 21812  df-asp 21813  df-ascl 21814  df-psr 21869  df-mvr 21870  df-mpl 21871  df-opsr 21873  df-evls 22033  df-evl 22034  df-psr1 22124  df-ply1 22126  df-evl1 22264

This theorem is referenced by:  pf1ind  22303