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Theorem mvrid 22022

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 12540	. . . 4 ⊢ 1 ∈ ℕ₀
9	2	snifpsrbag 21958	. . . 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 22021	. 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 4538	. 2 ⊢ if((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = 1
14	11, 13	eqtrdi 2791	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ifcif 4531 ↦ cmpt 5231 ^◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 ℕcn 12264 ℕ₀cn0 12524 0_gc0g 17486 1_rcur 20199 mVar cmvr 21943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-nn 12265 df-n0 12525 df-mvr 21948

This theorem is referenced by: mvrf1 22024

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