Theorem mvrfval 22201

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)

Assertion

Ref	Expression
mvrfval	⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))

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

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

Proof of Theorem mvrfval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrfval.v	. 2 ⊢ 𝑉 = (𝐼 mVar 𝑅)
2		mvrfval.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3	2	elexd 3476	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mvrfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
5	4	elexd 3476	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6	2	mptexd 7227	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V)
7		simpl 488	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
8	7	oveq2d 7433	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
9	8	rabeqdv 3429	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
10		mvrfval.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
11	9, 10	eqtr4di 2815	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
12		mpteq1 5198	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
14	13	eqeq2d 2773	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))
15		simpr 490	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
16	15	fveq2d 6886	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (1r‘𝑟) = (1r‘𝑅))
17		mvrfval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
18	16, 17	eqtr4di 2815	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (1r‘𝑟) = 1 )
19	15	fveq2d 6886	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = (0g‘𝑅))
20		mvrfval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
21	19, 20	eqtr4di 2815	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = 0 )
22	14, 18, 21	ifbieq12d 4514	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟)) = if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))
23	11, 22	mpteq12dv 5196	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟))) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))
24	7, 23	mpteq12dv 5196	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟)))) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
25		df-mvr 22131	. . . 4 ⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟)))))
26	24, 25	ovmpoga 7571	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V) → (𝐼 mVar 𝑅) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
27	3, 5, 6, 26	syl3anc 1398	. 2 ⊢ (𝜑 → (𝐼 mVar 𝑅) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
28	1, 27	eqtrid 2809	1 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  ifcif 4485   ↦ cmpt 5190  ^◡ccnv 5658   “ cima 5662  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830  Fincfn 8956  0cc0 11128  1c1 11129  ℕcn 12261  ℕ₀cn0 12532  0_gc0g 17530  1_rcur 20326   mVar cmvr 22126

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7420  df-oprab 7421  df-mpo 7422  df-mvr 22131

This theorem is used by:  mvrval  22202  mvrf  22205  subrgmvr  22255  esplyfvaln  34092