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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 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
mvrfval.z 0 = (0g𝑅)
mvrfval.o 1 = (1r𝑅)
mvrfval.i (𝜑𝐼𝑊)
mvrfval.r (𝜑𝑅𝑌)
Assertion
Ref Expression
mvrfval (𝜑𝑉 = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
Distinct variable groups:   𝑥,𝑓, 0   1 ,𝑓,𝑥   𝑦,𝑓,𝐷,𝑥   𝑦,𝑊   𝑓,,𝐼,𝑥,𝑦   𝑅,𝑓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,)   𝐷()   𝑅(𝑦,)   1 (𝑦,)   𝑉(𝑥,𝑦,𝑓,)   𝑊(𝑥,𝑓,)   𝑌(𝑥,𝑦,𝑓,)   0 (𝑦,)

Proof of Theorem mvrfval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvrfval.v . 2 𝑉 = (𝐼 mVar 𝑅)
2 mvrfval.i . . . 4 (𝜑𝐼𝑊)
32elexd 3465 . . 3 (𝜑𝐼 ∈ V)
4 mvrfval.r . . . 4 (𝜑𝑅𝑌)
54elexd 3465 . . 3 (𝜑𝑅 ∈ V)
62mptexd 7174 . . 3 (𝜑 → (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V)
7 simpl 483 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → 𝑖 = 𝐼)
87oveq2d 7373 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
98rabeqdv 3422 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
10 mvrfval.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
119, 10eqtr4di 2794 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
12 mpteq1 5198 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
1312adantr 481 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
1413eqeq2d 2747 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))
15 simpr 485 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → 𝑟 = 𝑅)
1615fveq2d 6846 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (1r𝑟) = (1r𝑅))
17 mvrfval.o . . . . . . . 8 1 = (1r𝑅)
1816, 17eqtr4di 2794 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (1r𝑟) = 1 )
1915fveq2d 6846 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (0g𝑟) = (0g𝑅))
20 mvrfval.z . . . . . . . 8 0 = (0g𝑅)
2119, 20eqtr4di 2794 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (0g𝑟) = 0 )
2214, 18, 21ifbieq12d 4514 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)) = if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))
2311, 22mpteq12dv 5196 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟))) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))
247, 23mpteq12dv 5196 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
25 df-mvr 21312 . . . 4 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
2624, 25ovmpoga 7509 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V) → (𝐼 mVar 𝑅) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
273, 5, 6, 26syl3anc 1371 . 2 (𝜑 → (𝐼 mVar 𝑅) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
281, 27eqtrid 2788 1 (𝜑𝑉 = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
Colors of variables: wff setvar class
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  cn 12153  0cn0 12413  0gc0g 17321  1rcur 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
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