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Theorem mvrid 21944

Description: The 𝑋𝑖-th coefficient of the term 𝑋𝑖 is 1. (Contributed by Mario Carneiro, 7-Jan-2015.)

**Hypotheses**
Ref	Expression
mvrfval.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
mvrfval.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
mvrfval.z	⊢ 0 = (0_g‘𝑅)
mvrfval.o	⊢ 1 = (1_r‘𝑅)
mvrfval.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mvrfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑌)
mvrval.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)

**Assertion**
Ref	Expression
mvrid	⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )

Distinct variable groups: 𝑦,𝐷 𝑦,𝑊 𝑦,ℎ,𝐼 ℎ,𝑋,𝑦 Allowed substitution hints: 𝜑(𝑦,ℎ) 𝐷(ℎ) 𝑅(𝑦,ℎ) 1 (𝑦,ℎ) 𝑉(𝑦,ℎ) 𝑊(ℎ) 𝑌(𝑦,ℎ) 0 (𝑦,ℎ)

**Proof of Theorem mvrid**
Step	Hyp	Ref	Expression
1		mvrfval.v	. . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅)
2		mvrfval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
3		mvrfval.z	. . 3 ⊢ 0 = (0_g‘𝑅)
4		mvrfval.o	. . 3 ⊢ 1 = (1_r‘𝑅)
5		mvrfval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
6		mvrfval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
7		mvrval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
8		1nn0 12517	. . . 4 ⊢ 1 ∈ ℕ₀
9	2	snifpsrbag 21880	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ₀) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ 𝐷)
10	5, 8, 9	sylancl 586	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ 𝐷)
11	1, 2, 3, 4, 5, 6, 7, 10	mvrval2 21943	. 2 ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = if((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
12		eqid 2735	. . 3 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
13	12	iftruei 4507	. 2 ⊢ if((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = 1
14	11, 13	eqtrdi 2786	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3415 ifcif 4500 ↦ cmpt 5201 ^◡ccnv 5653 “ cima 5657 ‘cfv 6531 (class class class)co 7405 ↑_m cmap 8840 Fincfn 8959 0cc0 11129 1c1 11130 ℕcn 12240 ℕ₀cn0 12501 0_gc0g 17453 1_rcur 20141 mVar cmvr 21865

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-nn 12241 df-n0 12502 df-mvr 21870

This theorem is referenced by: mvrf1 21946

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