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Theorem mvrval 22020
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)
mvrval.x (𝜑𝑋𝐼)
Assertion
Ref Expression
mvrval (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
Distinct variable groups:   0 ,𝑓   1 ,𝑓   𝑦,𝑓,𝐷   𝑦,𝑊   𝑓,,𝐼,𝑦   𝑅,𝑓   𝑓,𝑋,,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑓,)   𝐷()   𝑅(𝑦,)   1 (𝑦,)   𝑉(𝑦,𝑓,)   𝑊(𝑓,)   𝑌(𝑦,𝑓,)   0 (𝑦,)

Proof of Theorem mvrval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mvrfval.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
2 mvrfval.d . . . 4 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
3 mvrfval.z . . . 4 0 = (0g𝑅)
4 mvrfval.o . . . 4 1 = (1r𝑅)
5 mvrfval.i . . . 4 (𝜑𝐼𝑊)
6 mvrfval.r . . . 4 (𝜑𝑅𝑌)
71, 2, 3, 4, 5, 6mvrfval 22019 . . 3 (𝜑𝑉 = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
87fveq1d 6909 . 2 (𝜑 → (𝑉𝑋) = ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋))
9 mvrval.x . . 3 (𝜑𝑋𝐼)
10 eqeq2 2747 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 = 𝑥𝑦 = 𝑋))
1110ifbid 4554 . . . . . . . 8 (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
1211mpteq2dv 5250 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
1312eqeq2d 2746 . . . . . 6 (𝑥 = 𝑋 → (𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1413ifbid 4554 . . . . 5 (𝑥 = 𝑋 → if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ) = if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
1514mpteq2dv 5250 . . . 4 (𝑥 = 𝑋 → (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
16 eqid 2735 . . . 4 (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))
17 ovex 7464 . . . . . 6 (ℕ0m 𝐼) ∈ V
182, 17rabex2 5347 . . . . 5 𝐷 ∈ V
1918mptex 7243 . . . 4 (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V
2015, 16, 19fvmpt 7016 . . 3 (𝑋𝐼 → ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
219, 20syl 17 . 2 (𝜑 → ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
228, 21eqtrd 2775 1 (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  ifcif 4531  cmpt 5231  ccnv 5688  cima 5692  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  0cc0 11153  1c1 11154  cn 12264  0cn0 12524  0gc0g 17486  1rcur 20199   mVar cmvr 21943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mvr 21948
This theorem is referenced by:  mvrval2  22021  mplcoe3  22074  evlslem1  22124  rhmply1vr1  22407
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