Theorem mvrfval 21389

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 3465	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mvrfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
5	4	elexd 3465	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6	2	mptexd 7174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V)
7		simpl 483	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
8	7	oveq2d 7373	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
9	8	rabeqdv 3422	. . . . . . 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 2794	. . . . . 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 481	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
14	13	eqeq2d 2747	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))
15		simpr 485	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
16	15	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (1r‘𝑟) = (1r‘𝑅))
17		mvrfval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
18	16, 17	eqtr4di 2794	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (1r‘𝑟) = 1 )
19	15	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = (0g‘𝑅))
20		mvrfval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
21	19, 20	eqtr4di 2794	. . . . . . 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 21312	. . . 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 7509	. . 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 1371	. 2 ⊢ (𝜑 → (𝐼 mVar 𝑅) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
28	1, 27	eqtrid 2788	1 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3407  Vcvv 3445  ifcif 4486   ↦ cmpt 5188  ^◡ccnv 5632   “ cima 5636  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052  ℕcn 12153  ℕ₀cn0 12413  0_gc0g 17321  1_rcur 19913   mVar cmvr 21307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-mvr 21312

This theorem is referenced by:  mvrval  21390  mvrf  21393  subrgmvr  21434