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Theorem mvrfval 21540
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 3495 . . 3 (𝜑𝐼 ∈ V)
4 mvrfval.r . . . 4 (𝜑𝑅𝑌)
54elexd 3495 . . 3 (𝜑𝑅 ∈ V)
62mptexd 7226 . . 3 (𝜑 → (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V)
7 simpl 484 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → 𝑖 = 𝐼)
87oveq2d 7425 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
98rabeqdv 3448 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
10 mvrfval.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
119, 10eqtr4di 2791 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
12 mpteq1 5242 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
1312adantr 482 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
1413eqeq2d 2744 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))
15 simpr 486 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → 𝑟 = 𝑅)
1615fveq2d 6896 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (1r𝑟) = (1r𝑅))
17 mvrfval.o . . . . . . . 8 1 = (1r𝑅)
1816, 17eqtr4di 2791 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (1r𝑟) = 1 )
1915fveq2d 6896 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (0g𝑟) = (0g𝑅))
20 mvrfval.z . . . . . . . 8 0 = (0g𝑅)
2119, 20eqtr4di 2791 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (0g𝑟) = 0 )
2214, 18, 21ifbieq12d 4557 . . . . . 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 5240 . . . . 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 5240 . . . 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 21463 . . . 4 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
2624, 25ovmpoga 7562 . . 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 1372 . 2 (𝜑 → (𝐼 mVar 𝑅) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
281, 27eqtrid 2785 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 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  ifcif 4529  cmpt 5232  ccnv 5676  cima 5680  cfv 6544  (class class class)co 7409  m cmap 8820  Fincfn 8939  0cc0 11110  1c1 11111  cn 12212  0cn0 12472  0gc0g 17385  1rcur 20004   mVar cmvr 21458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-mvr 21463
This theorem is referenced by:  mvrval  21541  mvrf  21544  subrgmvr  21588
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