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Theorem mvrid 21926

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 12519	. . . 4 ⊢ 1 ∈ ℕ₀
9	2	snifpsrbag 21855	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ₀) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ 𝐷)
10	5, 8, 9	sylancl 585	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ 𝐷)
11	1, 2, 3, 4, 5, 6, 7, 10	mvrval2 21925	. 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 2728	. . 3 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
13	12	iftruei 4536	. 2 ⊢ if((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = 1
14	11, 13	eqtrdi 2784	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3429 ifcif 4529 ↦ cmpt 5231 ^◡ccnv 5677 “ cima 5681 ‘cfv 6548 (class class class)co 7420 ↑_m cmap 8845 Fincfn 8964 0cc0 11139 1c1 11140 ℕcn 12243 ℕ₀cn0 12503 0_gc0g 17421 1_rcur 20121 mVar cmvr 21838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-nn 12244 df-n0 12504 df-mvr 21843

This theorem is referenced by: mvrf1 21928

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