Theorem mvrfval 21546

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 3494	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mvrfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
5	4	elexd 3494	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6	2	mptexd 7228	. . 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 7427	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
9	8	rabeqdv 3447	. . . . . . 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 2790	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
12		mpteq1 5241	. . . . . . . . 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 2743	. . . . . . 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 6895	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (1r‘𝑟) = (1r‘𝑅))
17		mvrfval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
18	16, 17	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (1r‘𝑟) = 1 )
19	15	fveq2d 6895	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = (0g‘𝑅))
20		mvrfval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
21	19, 20	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (0g‘𝑟) = 0 )
22	14, 18, 21	ifbieq12d 4556	. . . . . 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 5239	. . . . 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 5239	. . . 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 21469	. . . 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 7564	. . 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 2784	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 3432  Vcvv 3474  ifcif 4528   ↦ cmpt 5231  ^◡ccnv 5675   “ cima 5679  ‘cfv 6543  (class class class)co 7411   ↑_m cmap 8822  Fincfn 8941  0cc0 11112  1c1 11113  ℕcn 12214  ℕ₀cn0 12474  0_gc0g 17387  1_rcur 20006   mVar cmvr 21464

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-mvr 21469

This theorem is referenced by:  mvrval  21547  mvrf  21550  subrgmvr  21594