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Theorem mvrid 22004

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 12542	. . . 4 ⊢ 1 ∈ ℕ₀
9	2	snifpsrbag 21940	. . . 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 22003	. 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 2737	. . 3 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
13	12	iftruei 4532	. 2 ⊢ if((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = 1
14	11, 13	eqtrdi 2793	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 3436 ifcif 4525 ↦ cmpt 5225 ^◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 Fincfn 8985 0cc0 11155 1c1 11156 ℕcn 12266 ℕ₀cn0 12526 0_gc0g 17484 1_rcur 20178 mVar cmvr 21925

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-nn 12267 df-n0 12527 df-mvr 21930

This theorem is referenced by: mvrf1 22006

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