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Theorem mplmon2 22283

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 2762	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
5		eqid 2762	. . 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 22257	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃))
14	1, 2, 3, 4, 5, 6, 7, 13	mplvsca 22235	. 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))))
15		ovex 7450	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	6, 15	rabex2 5309	. . . 4 ⊢ 𝐷 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
18	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵)
19	9	fvexi 6896	. . . . 5 ⊢ 1 ∈ V
20	8	fvexi 6896	. . . . 5 ⊢ 0 ∈ V
21	19, 20	ifex 4536	. . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V)
23		fconstmpt 5721	. . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)
24	23	a1i 11	. . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋))
25		eqidd 2763	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))
26	17, 18, 22, 24, 25	offval2 7702	. 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 7425	. . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 1 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
28	27	eqeq1d 2764	. . . 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 7425	. . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 0 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
30	29	eqeq1d 2764	. . . 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 20417	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 1 ) = 𝑋)
32	11, 7, 31	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 1 ) = 𝑋)
33		iftrue 4491	. . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋)
34	33	eqcomd 2768	. . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 ))
35	32, 34	sylan9eq 2817	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
36	3, 5, 8	ringrz 20442	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 0 ) = 0 )
37	11, 7, 36	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 0 ) = 0 )
38		iffalse 4494	. . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 )
39	38	eqcomd 2768	. . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 ))
40	37, 39	sylan9eq 2817	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
41	28, 30, 35, 40	ifbothda 4524	. . 3 ⊢ (𝜑 → (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 ))
42	41	mpteq2dv 5203	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))
43	14, 26, 42	3eqtrd 2801	1 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3414 Vcvv 3453 ifcif 4485 {csn 4587 ↦ cmpt 5190 × cxp 5657 ^◡ccnv 5658 “ cima 5662 ‘cfv 6537 (class class class)co 7417 ∘_f cof 7680 ↑_m cmap 8830 Fincfn 8956 ℕcn 12261 ℕ₀cn0 12532 Basecbs 17307 ._rcmulr 17349 ·_𝑠 cvsca 17352 0_gc0g 17530 1_rcur 20326 Ringcrg 20378 mPoly cmpl 22127

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-tset 17367 df-0g 17532 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-grp 19066 df-minusg 19067 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-psr 22130 df-mpl 22132

This theorem is used by: mplascl 22286 mplmon2cl 22290 mplmon2mul 22291 mplcoe4 22293 coe1tm 22505

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