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Theorem mplmon2 21179

Description: Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)

**Hypotheses**
Ref	Expression
mplmon2.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplmon2.v	⊢ · = ( ·_𝑠 ‘𝑃)
mplmon2.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
mplmon2.o	⊢ 1 = (1_r‘𝑅)
mplmon2.z	⊢ 0 = (0_g‘𝑅)
mplmon2.b	⊢ 𝐵 = (Base‘𝑅)
mplmon2.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplmon2.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplmon2.k	⊢ (𝜑 → 𝐾 ∈ 𝐷)
mplmon2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
mplmon2	⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Distinct variable groups: 𝜑,𝑦 𝑦,𝐵 𝑦,𝐷 𝑓,𝐼 𝑓,𝐾,𝑦 𝑦, 1 𝑦,𝑅 𝑦,𝑋 𝑦, 0 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐷(𝑓) 𝑃(𝑦,𝑓) 𝑅(𝑓) · (𝑦,𝑓) 1 (𝑓) 𝐼(𝑦) 𝑊(𝑦,𝑓) 𝑋(𝑓) 0 (𝑓)

**Proof of Theorem mplmon2**
Step	Hyp	Ref	Expression
1		mplmon2.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		mplmon2.v	. . 3 ⊢ · = ( ·_𝑠 ‘𝑃)
3		mplmon2.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2738	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
5		eqid 2738	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
6		mplmon2.d	. . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
7		mplmon2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		mplmon2.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
9		mplmon2.o	. . . 4 ⊢ 1 = (1_r‘𝑅)
10		mplmon2.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
11		mplmon2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
12		mplmon2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
13	1, 4, 8, 9, 6, 10, 11, 12	mplmon 21146	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃))
14	1, 2, 3, 4, 5, 6, 7, 13	mplvsca 21129	. 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))))
15		ovex 7288	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	6, 15	rabex2 5253	. . . 4 ⊢ 𝐷 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
18	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵)
19	9	fvexi 6770	. . . . 5 ⊢ 1 ∈ V
20	8	fvexi 6770	. . . . 5 ⊢ 0 ∈ V
21	19, 20	ifex 4506	. . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V)
23		fconstmpt 5640	. . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)
24	23	a1i 11	. . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋))
25		eqidd 2739	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))
26	17, 18, 22, 24, 25	offval2 7531	. 2 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))))
27		oveq2 7263	. . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 1 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
28	27	eqeq1d 2740	. . . 4 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → ((𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ) ↔ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )))
29		oveq2 7263	. . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 0 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
30	29	eqeq1d 2740	. . . 4 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → ((𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ) ↔ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )))
31	3, 5, 9	ringridm 19726	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 1 ) = 𝑋)
32	11, 7, 31	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 1 ) = 𝑋)
33		iftrue 4462	. . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋)
34	33	eqcomd 2744	. . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 ))
35	32, 34	sylan9eq 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
36	3, 5, 8	ringrz 19742	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 0 ) = 0 )
37	11, 7, 36	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 0 ) = 0 )
38		iffalse 4465	. . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 )
39	38	eqcomd 2744	. . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 ))
40	37, 39	sylan9eq 2799	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
41	28, 30, 35, 40	ifbothda 4494	. . 3 ⊢ (𝜑 → (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 ))
42	41	mpteq2dv 5172	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))
43	14, 26, 42	3eqtrd 2782	1 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 ifcif 4456 {csn 4558 ↦ cmpt 5153 × cxp 5578 ^◡ccnv 5579 “ cima 5583 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 ↑_m cmap 8573 Fincfn 8691 ℕcn 11903 ℕ₀cn0 12163 Basecbs 16840 ._rcmulr 16889 ·_𝑠 cvsca 16892 0_gc0g 17067 1_rcur 19652 Ringcrg 19698 mPoly cmpl 21019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-tset 16907 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-mgp 19636 df-ur 19653 df-ring 19700 df-psr 21022 df-mpl 21024

This theorem is referenced by: mplascl 21182 mplmon2cl 21186 mplmon2mul 21187 mplcoe4 21189 coe1tm 21354

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