mpfpf1 - Metamath Proof Explorer

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Theorem mpfpf1 21869

Description: Convert a multivariate polynomial function to univariate. (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 (1_o eval 𝑅)

**Assertion**
Ref	Expression
mpfpf1	⊢ (𝐹 ∈ 𝐸 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄)

Distinct variable groups: 𝑦,𝐵 𝑦,𝐸 𝑦,𝐹 𝑦,𝑅 Allowed substitution hint: 𝑄(𝑦)

Proof of Theorem mpfpf1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpfpf1.q	. . . . 5 ⊢ 𝐸 = ran (1_o eval 𝑅)
2		eqid 2732	. . . . . . 7 ⊢ (1_o eval 𝑅) = (1_o eval 𝑅)
3		pf1f.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	evlval 21657	. . . . . 6 ⊢ (1_o eval 𝑅) = ((1_o evalSub 𝑅)‘𝐵)
5	4	rneqi 5936	. . . . 5 ⊢ ran (1_o eval 𝑅) = ran ((1_o evalSub 𝑅)‘𝐵)
6	1, 5	eqtri 2760	. . . 4 ⊢ 𝐸 = ran ((1_o evalSub 𝑅)‘𝐵)
7	6	mpfrcl 21647	. . 3 ⊢ (𝐹 ∈ 𝐸 → (1_o ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅)))
8	7	simp2d 1143	. 2 ⊢ (𝐹 ∈ 𝐸 → 𝑅 ∈ CRing)
9		id 22	. . . 4 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ 𝐸)
10	9, 1	eleqtrdi 2843	. . 3 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ ran (1_o eval 𝑅))
11		1on 8477	. . . . 5 ⊢ 1_o ∈ On
12		eqid 2732	. . . . . 6 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
13		eqid 2732	. . . . . 6 ⊢ (𝑅 ↑_s (𝐵 ↑_m 1_o)) = (𝑅 ↑_s (𝐵 ↑_m 1_o))
14	2, 3, 12, 13	evlrhm 21658	. . . . 5 ⊢ ((1_o ∈ On ∧ 𝑅 ∈ CRing) → (1_o eval 𝑅) ∈ ((1_o mPoly 𝑅) RingHom (𝑅 ↑_s (𝐵 ↑_m 1_o))))
15	11, 8, 14	sylancr 587	. . . 4 ⊢ (𝐹 ∈ 𝐸 → (1_o eval 𝑅) ∈ ((1_o mPoly 𝑅) RingHom (𝑅 ↑_s (𝐵 ↑_m 1_o))))
16		eqid 2732	. . . . . 6 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
17		eqid 2732	. . . . . 6 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
18		eqid 2732	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
19	16, 17, 18	ply1bas 21718	. . . . 5 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
20		eqid 2732	. . . . 5 ⊢ (Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o))) = (Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o)))
21	19, 20	rhmf 20262	. . . 4 ⊢ ((1_o eval 𝑅) ∈ ((1_o mPoly 𝑅) RingHom (𝑅 ↑_s (𝐵 ↑_m 1_o))) → (1_o eval 𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o))))
22		ffn 6717	. . . 4 ⊢ ((1_o eval 𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o))) → (1_o eval 𝑅) Fn (Base‘(Poly₁‘𝑅)))
23		fvelrnb 6952	. . . 4 ⊢ ((1_o eval 𝑅) Fn (Base‘(Poly₁‘𝑅)) → (𝐹 ∈ ran (1_o eval 𝑅) ↔ ∃𝑥 ∈ (Base‘(Poly₁‘𝑅))((1_o eval 𝑅)‘𝑥) = 𝐹))
24	15, 21, 22, 23	4syl 19	. . 3 ⊢ (𝐹 ∈ 𝐸 → (𝐹 ∈ ran (1_o eval 𝑅) ↔ ∃𝑥 ∈ (Base‘(Poly₁‘𝑅))((1_o eval 𝑅)‘𝑥) = 𝐹))
25	10, 24	mpbid 231	. 2 ⊢ (𝐹 ∈ 𝐸 → ∃𝑥 ∈ (Base‘(Poly₁‘𝑅))((1_o eval 𝑅)‘𝑥) = 𝐹)
26		eqid 2732	. . . . . 6 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
27	26, 2, 3, 12, 19	evl1val 21847	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) = (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
28		eqid 2732	. . . . . . . . 9 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
29	26, 16, 28, 3	evl1rhm 21850	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
30		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
31	18, 30	rhmf 20262	. . . . . . . 8 ⊢ ((eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
32		ffn 6717	. . . . . . . 8 ⊢ ((eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)) → (eval₁‘𝑅) Fn (Base‘(Poly₁‘𝑅)))
33	29, 31, 32	3syl 18	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) Fn (Base‘(Poly₁‘𝑅)))
34		fnfvelrn 7082	. . . . . . 7 ⊢ (((eval₁‘𝑅) Fn (Base‘(Poly₁‘𝑅)) ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) ∈ ran (eval₁‘𝑅))
35	33, 34	sylan 580	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) ∈ ran (eval₁‘𝑅))
36		pf1rcl.q	. . . . . 6 ⊢ 𝑄 = ran (eval₁‘𝑅)
37	35, 36	eleqtrrdi 2844	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) ∈ 𝑄)
38	27, 37	eqeltrrd 2834	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄)
39		coeq1 5857	. . . . 5 ⊢ (((1_o eval 𝑅)‘𝑥) = 𝐹 → (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) = (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
40	39	eleq1d 2818	. . . 4 ⊢ (((1_o eval 𝑅)‘𝑥) = 𝐹 → ((((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄 ↔ (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄))
41	38, 40	syl5ibcom 244	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → (((1_o eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄))
42	41	rexlimdva 3155	. 2 ⊢ (𝑅 ∈ CRing → (∃𝑥 ∈ (Base‘(Poly₁‘𝑅))((1_o eval 𝑅)‘𝑥) = 𝐹 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄))
43	8, 25, 42	sylc 65	1 ⊢ (𝐹 ∈ 𝐸 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 {csn 4628 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ∘ ccom 5680 Oncon0 6364 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 1_oc1o 8458 ↑_m cmap 8819 Basecbs 17143 ↑_s cpws 17391 CRingccrg 20056 RingHom crh 20247 SubRingcsubrg 20314 mPoly cmpl 21458 evalSub ces 21632 eval cevl 21633 PwSer₁cps1 21698 Poly₁cpl1 21700 eval₁ce1 21832

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-srg 20009 df-ring 20057 df-cring 20058 df-rnghom 20250 df-subrg 20316 df-lmod 20472 df-lss 20542 df-lsp 20582 df-assa 21407 df-asp 21408 df-ascl 21409 df-psr 21461 df-mvr 21462 df-mpl 21463 df-opsr 21465 df-evls 21634 df-evl 21635 df-psr1 21703 df-ply1 21705 df-evl1 21834

This theorem is referenced by: pf1ind 21873

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