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Theorem mvrfval 19642
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)
3 elex 3352 . . . 4 (𝐼𝑊𝐼 ∈ V)
42, 3syl 17 . . 3 (𝜑𝐼 ∈ V)
5 mvrfval.r . . . 4 (𝜑𝑅𝑌)
6 elex 3352 . . . 4 (𝑅𝑌𝑅 ∈ V)
75, 6syl 17 . . 3 (𝜑𝑅 ∈ V)
8 mptexg 6649 . . . 4 (𝐼𝑊 → (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V)
92, 8syl 17 . . 3 (𝜑 → (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) ∈ V)
10 simpl 474 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → 𝑖 = 𝐼)
1110oveq2d 6830 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (ℕ0𝑚 𝑖) = (ℕ0𝑚 𝐼))
12 rabeq 3332 . . . . . . . 8 ((ℕ0𝑚 𝑖) = (ℕ0𝑚 𝐼) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin})
1311, 12syl 17 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin})
14 mvrfval.d . . . . . . 7 𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
1513, 14syl6eqr 2812 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
16 mpteq1 4889 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
1716adantr 472 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
1817eqeq2d 2770 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))
19 simpr 479 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → 𝑟 = 𝑅)
2019fveq2d 6357 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (1r𝑟) = (1r𝑅))
21 mvrfval.o . . . . . . . 8 1 = (1r𝑅)
2220, 21syl6eqr 2812 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (1r𝑟) = 1 )
2319fveq2d 6357 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (0g𝑟) = (0g𝑅))
24 mvrfval.z . . . . . . . 8 0 = (0g𝑅)
2523, 24syl6eqr 2812 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (0g𝑟) = 0 )
2618, 22, 25ifbieq12d 4257 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)) = if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))
2715, 26mpteq12dv 4885 . . . . 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 )))
2810, 27mpteq12dv 4885 . . . 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 ))))
29 df-mvr 19579 . . . 4 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
3028, 29ovmpt2ga 6956 . . 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 ))))
314, 7, 9, 30syl3anc 1477 . 2 (𝜑 → (𝐼 mVar 𝑅) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
321, 31syl5eq 2806 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 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  ifcif 4230  cmpt 4881  ccnv 5265  cima 5269  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  Fincfn 8123  0cc0 10148  1c1 10149  cn 11232  0cn0 11504  0gc0g 16322  1rcur 18721   mVar cmvr 19574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-mvr 19579
This theorem is referenced by:  mvrval  19643  mvrf  19646  subrgmvr  19683
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