mpfpf1 - Metamath Proof Explorer

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Theorem mpfpf1 22270

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 2728	. . . . . . 7 ⊢ (1_o eval 𝑅) = (1_o eval 𝑅)
3		pf1f.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	evlval 22041	. . . . . 6 ⊢ (1_o eval 𝑅) = ((1_o evalSub 𝑅)‘𝐵)
5	4	rneqi 5939	. . . . 5 ⊢ ran (1_o eval 𝑅) = ran ((1_o evalSub 𝑅)‘𝐵)
6	1, 5	eqtri 2756	. . . 4 ⊢ 𝐸 = ran ((1_o evalSub 𝑅)‘𝐵)
7	6	mpfrcl 22031	. . 3 ⊢ (𝐹 ∈ 𝐸 → (1_o ∈ V ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅)))
8	7	simp2d 1141	. 2 ⊢ (𝐹 ∈ 𝐸 → 𝑅 ∈ CRing)
9		id 22	. . . 4 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ 𝐸)
10	9, 1	eleqtrdi 2839	. . 3 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ ran (1_o eval 𝑅))
11		1on 8499	. . . . 5 ⊢ 1_o ∈ On
12		eqid 2728	. . . . . 6 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
13		eqid 2728	. . . . . 6 ⊢ (𝑅 ↑_s (𝐵 ↑_m 1_o)) = (𝑅 ↑_s (𝐵 ↑_m 1_o))
14	2, 3, 12, 13	evlrhm 22042	. . . . 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 2728	. . . . . 6 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
17		eqid 2728	. . . . . 6 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
18		eqid 2728	. . . . . 6 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
19	16, 17, 18	ply1bas 22114	. . . . 5 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
20		eqid 2728	. . . . 5 ⊢ (Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o))) = (Base‘(𝑅 ↑_s (𝐵 ↑_m 1_o)))
21	19, 20	rhmf 20424	. . . 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 6722	. . . 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 6959	. . . 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 2728	. . . . . 6 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
27	26, 2, 3, 12, 19	evl1val 22248	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) = (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))))
28		eqid 2728	. . . . . . . . 9 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
29	26, 16, 28, 3	evl1rhm 22251	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)))
30		eqid 2728	. . . . . . . . 9 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
31	18, 30	rhmf 20424	. . . . . . . 8 ⊢ ((eval₁‘𝑅) ∈ ((Poly₁‘𝑅) RingHom (𝑅 ↑_s 𝐵)) → (eval₁‘𝑅):(Base‘(Poly₁‘𝑅))⟶(Base‘(𝑅 ↑_s 𝐵)))
32		ffn 6722	. . . . . . . 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 7090	. . . . . . 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 2840	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → ((eval₁‘𝑅)‘𝑥) ∈ 𝑄)
38	27, 37	eqeltrrd 2830	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (Base‘(Poly₁‘𝑅))) → (((1_o eval 𝑅)‘𝑥) ∘ (𝑦 ∈ 𝐵 ↦ (1_o × {𝑦}))) ∈ 𝑄)
39		coeq1 5860	. . . . 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 2814	. . . 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 3152	. 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 1534 ∈ wcel 2099 ∃wrex 3067 Vcvv 3471 {csn 4629 ↦ cmpt 5231 × cxp 5676 ran crn 5679 ∘ ccom 5682 Oncon0 6369 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 1_oc1o 8480 ↑_m cmap 8845 Basecbs 17180 ↑_s cpws 17428 CRingccrg 20174 RingHom crh 20408 SubRingcsubrg 20506 mPoly cmpl 21839 evalSub ces 22016 eval cevl 22017 PwSer₁cps1 22094 Poly₁cpl1 22096 eval₁ce1 22233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-cring 20176 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-assa 21787 df-asp 21788 df-ascl 21789 df-psr 21842 df-mvr 21843 df-mpl 21844 df-opsr 21846 df-evls 22018 df-evl 22019 df-psr1 22099 df-ply1 22101 df-evl1 22235

This theorem is referenced by: pf1ind 22274

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