Theorem mvrval 22294

Description: Value of the generating elements of the power series structure. (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 = (0g‘𝑅)
mvrfval.o	⊢ 1 = (1r‘𝑅)
mvrfval.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mvrfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑌)
mvrval.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)

Assertion

Ref	Expression
mvrval	⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))

Distinct variable groups:   0 ,𝑓   1 ,𝑓   𝑦,𝑓,𝐷   𝑦,𝑊   𝑓,ℎ,𝐼,𝑦   𝑅,𝑓   𝑓,𝑋,ℎ,𝑦

Allowed substitution hints:   𝜑(𝑦, 𝑓, ℎ)   𝐷(ℎ)   𝑅(𝑦, ℎ)   1 (𝑦, ℎ)   𝑉(𝑦, 𝑓, ℎ)   𝑊(𝑓, ℎ)   𝑌(𝑦, 𝑓, ℎ)   0 (𝑦, ℎ)

Proof of Theorem mvrval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrfval.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
2		mvrfval.d	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
3		mvrfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
4		mvrfval.o	. . . 4 ⊢ 1 = (1r‘𝑅)
5		mvrfval.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
6		mvrfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
7	1, 2, 3, 4, 5, 6	mvrfval 22293	. . 3 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
8	7	fveq1d 6887	. 2 ⊢ (𝜑 → (𝑉‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋))
9		mvrval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
10		eqeq2 2772	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋))
11	10	ifbid 4505	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
12	11	mpteq2dv 5198	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
13	12	eqeq2d 2771	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
14	13	ifbid 4505	. . . . 5 ⊢ (𝑥 = 𝑋 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ) = if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
15	14	mpteq2dv 5198	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
16		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))
17		ovex 7453	. . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V
18	2, 17	rabex2 5301	. . . . 5 ⊢ 𝐷 ∈ V
19	18	mptex 7229	. . . 4 ⊢ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V
20	15, 16, 19	fvmpt 6993	. . 3 ⊢ (𝑋 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
21	9, 20	syl 18	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
22	8, 21	eqtrd 2795	1 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412  ifcif 4481   ↦ cmpt 5185  ^◡ccnv 5649   “ cima 5653  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8850  Fincfn 8976  0cc0 11203  1c1 11204  ℕcn 12338  ℕ₀cn0 12609  0_gc0g 17615  1_rcur 20412   mVar cmvr 22218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-mvr 22223

This theorem is used by:  mvrval2  22295  mplcoe3  22352  evlslem1  22396  rhmply1vr1  22707