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Theorem mplmon2 22172

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 2765	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
5		eqid 2765	. . 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 22146	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃))
14	1, 2, 3, 4, 5, 6, 7, 13	mplvsca 22124	. 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))))
15		ovex 7433	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	6, 15	rabex2 5302	. . . 4 ⊢ 𝐷 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
18	7	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵)
19	9	fvexi 6885	. . . . 5 ⊢ 1 ∈ V
20	8	fvexi 6885	. . . . 5 ⊢ 0 ∈ V
21	19, 20	ifex 4534	. . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V)
23		fconstmpt 5714	. . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)
24	23	a1i 11	. . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋))
25		eqidd 2766	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))
26	17, 18, 22, 24, 25	offval2 7684	. 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 7408	. . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 1 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
28	27	eqeq1d 2767	. . . 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 7408	. . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 0 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
30	29	eqeq1d 2767	. . . 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 20344	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 1 ) = 𝑋)
32	11, 7, 31	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 1 ) = 𝑋)
33		iftrue 4489	. . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋)
34	33	eqcomd 2771	. . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 ))
35	32, 34	sylan9eq 2820	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
36	3, 5, 8	ringrz 20368	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 0 ) = 0 )
37	11, 7, 36	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 0 ) = 0 )
38		iffalse 4492	. . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 )
39	38	eqcomd 2771	. . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 ))
40	37, 39	sylan9eq 2820	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
41	28, 30, 35, 40	ifbothda 4522	. . 3 ⊢ (𝜑 → (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 ))
42	41	mpteq2dv 5199	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))
43	14, 26, 42	3eqtrd 2804	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 400 = wceq 1563 ∈ wcel 2145 {crab 3417 Vcvv 3457 ifcif 4483 {csn 4585 ↦ cmpt 5186 × cxp 5650 ^◡ccnv 5651 “ cima 5655 ‘cfv 6525 (class class class)co 7400 ∘_f cof 7662 ↑_m cmap 8812 Fincfn 8931 ℕcn 12224 ℕ₀cn0 12495 Basecbs 17259 ._rcmulr 17301 ·_𝑠 cvsca 17304 0_gc0g 17482 1_rcur 20254 Ringcrg 20306 mPoly cmpl 22016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-tset 17319 df-0g 17484 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-grp 18993 df-minusg 18994 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-psr 22019 df-mpl 22021

This theorem is referenced by: mplascl 22175 mplmon2cl 22179 mplmon2mul 22180 mplcoe4 22182 coe1tm 22394

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