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Theorem mvrid 22284

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 12615	. . . 4 ⊢ 1 ∈ ℕ₀
9	2	snifpsrbag 22221	. . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ₀) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ 𝐷)
10	5, 8, 9	sylancl 598	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ∈ 𝐷)
11	1, 2, 3, 4, 5, 6, 7, 10	mvrval2 22283	. 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 2761	. . 3 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
13	12	iftruei 4489	. 2 ⊢ if((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = 1
14	11, 13	eqtrdi 2812	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 {crab 3413 ifcif 4482 ↦ cmpt 5186 ^◡ccnv 5650 “ cima 5654 ‘cfv 6537 (class class class)co 7418 ↑_m cmap 8840 Fincfn 8966 0cc0 11193 1c1 11194 ℕcn 12328 ℕ₀cn0 12599 0_gc0g 17603 1_rcur 20400 mVar cmvr 22206

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 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269

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-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-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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-nn 12329 df-n0 12600 df-mvr 22211

This theorem is used by: mvrf1 22286

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