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Theorem mplmon2 22350

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 2761	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
5		eqid 2761	. . 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 22324	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃))
14	1, 2, 3, 4, 5, 6, 7, 13	mplvsca 22302	. 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))))
15		ovex 7445	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	6, 15	rabex2 5302	. . . 4 ⊢ 𝐷 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
18	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵)
19	9	fvexi 6891	. . . . 5 ⊢ 1 ∈ V
20	8	fvexi 6891	. . . . 5 ⊢ 0 ∈ V
21	19, 20	ifex 4533	. . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V)
23		fconstmpt 5713	. . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)
24	23	a1i 11	. . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋))
25		eqidd 2762	. . 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 7420	. . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 1 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
28	27	eqeq1d 2763	. . . 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 7420	. . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 0 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
30	29	eqeq1d 2763	. . . 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 20479	. . . . . 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 4488	. . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋)
34	33	eqcomd 2767	. . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 ))
35	32, 34	sylan9eq 2816	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
36	3, 5, 8	ringrz 20505	. . . . . 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 4491	. . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 )
39	38	eqcomd 2767	. . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 ))
40	37, 39	sylan9eq 2816	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
41	28, 30, 35, 40	ifbothda 4521	. . 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 2800	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 3413 Vcvv 3451 ifcif 4482 {csn 4584 ↦ cmpt 5186 × cxp 5649 ^◡ccnv 5650 “ cima 5654 ‘cfv 6531 (class class class)co 7412 ∘_f cof 7680 ↑_m cmap 8831 Fincfn 8957 ℕcn 12316 ℕ₀cn0 12587 Basecbs 17367 ._rcmulr 17409 ·_𝑠 cvsca 17412 0_gc0g 17590 1_rcur 20387 Ringcrg 20439 mPoly cmpl 22194

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-tset 17427 df-0g 17592 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-grp 19127 df-minusg 19128 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-psr 22197 df-mpl 22199

This theorem is used by: mplascl 22353 mplmon2cl 22357 mplmon2mul 22358 mplcoe4 22360 coe1tm 22572

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