mpfpf1 - Metamath Proof Explorer

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Theorem mpfpf1 22221

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 2726	. . . . . . 7 ⊢ (1_o eval 𝑅) = (1_o eval 𝑅)
3		pf1f.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	evlval 21996	. . . . . 6 ⊢ (1_o eval 𝑅) = ((1_o evalSub 𝑅)‘𝐵)
5	4	rneqi 5929	. . . . 5 ⊢ ran (1_o eval 𝑅) = ran ((1_o evalSub 𝑅)‘𝐵)
6	1, 5	eqtri 2754	. . . 4 ⊢ 𝐸 = ran ((1_o evalSub 𝑅)‘𝐵)
7	6	mpfrcl 21986	. . 3 ⊢ (𝐹 ∈ 𝐸 → (1_o ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅)))
8	7	simp2d 1140	. 2 ⊢ (𝐹 ∈ 𝐸 → 𝑅 ∈ CRing)
9		id 22	. . . 4 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ 𝐸)
10	9, 1	eleqtrdi 2837	. . 3 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ ran (1_o eval 𝑅))
11		1on 8476	. . . . 5 ⊢ 1_o ∈ On
12		eqid 2726	. . . . . 6 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
13		eqid 2726	. . . . . 6 ⊢ (𝑅 ↑_s (𝐵 ↑_m 1_o)) = (𝑅 ↑_s (𝐵 ↑_m 1_o))
14	2, 3, 12, 13	evlrhm 21997	. . . . 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 586	. . . 4 ⊢ (𝐹 ∈ 𝐸 → (1_o eval 𝑅) ∈ ((1_o mPoly 𝑅) RingHom (𝑅 ↑_s (𝐵 ↑_m 1_o))))
16		eqid 2726	. . . . . 6 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
17		eqid 2726	. . . . . 6 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
18		eqid 2726	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
19	16, 17, 18	ply1bas 22065	. . . . 5 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
20		eqid 2726	. . . . 5 ⊢ (Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o))) = (Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o)))
21	19, 20	rhmf 20385	. . . 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 6710	. . . 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 6945	. . . 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 2726	. . . . . 6 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
27	26, 2, 3, 12, 19	evl1val 22199	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) = (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
28		eqid 2726	. . . . . . . . 9 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
29	26, 16, 28, 3	evl1rhm 22202	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
30		eqid 2726	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
31	18, 30	rhmf 20385	. . . . . . . 8 ⊢ ((eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
32		ffn 6710	. . . . . . . 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 7075	. . . . . . 7 ⊢ (((eval₁‘𝑅) Fn (Base‘(Poly₁‘𝑅)) ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) ∈ ran (eval₁‘𝑅))
35	33, 34	sylan 579	. . . . . 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 2838	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) ∈ 𝑄)
38	27, 37	eqeltrrd 2828	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄)
39		coeq1 5850	. . . . 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 2812	. . . 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 3149	. 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 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 {csn 4623 ↦ cmpt 5224 × cxp 5667 ran crn 5670 ∘ ccom 5673 Oncon0 6357 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 1_oc1o 8457 ↑_m cmap 8819 Basecbs 17151 ↑_s cpws 17399 CRingccrg 20137 RingHom crh 20369 SubRingcsubrg 20467 mPoly cmpl 21796 evalSub ces 21971 eval cevl 21972 PwSer₁cps1 22045 Poly₁cpl1 22047 eval₁ce1 22184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-ghm 19137 df-cntz 19231 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-srg 20090 df-ring 20138 df-cring 20139 df-rhm 20372 df-subrng 20444 df-subrg 20469 df-lmod 20706 df-lss 20777 df-lsp 20817 df-assa 21744 df-asp 21745 df-ascl 21746 df-psr 21799 df-mvr 21800 df-mpl 21801 df-opsr 21803 df-evls 21973 df-evl 21974 df-psr1 22050 df-ply1 22052 df-evl1 22186

This theorem is referenced by: pf1ind 22225

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