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Theorem mvrval 21929
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 21928 . . 3 (𝜑𝑉 = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
87fveq1d 6902 . 2 (𝜑 → (𝑉𝑋) = ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋))
9 mvrval.x . . 3 (𝜑𝑋𝐼)
10 eqeq2 2739 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 = 𝑥𝑦 = 𝑋))
1110ifbid 4553 . . . . . . . 8 (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
1211mpteq2dv 5252 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
1312eqeq2d 2738 . . . . . 6 (𝑥 = 𝑋 → (𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1413ifbid 4553 . . . . 5 (𝑥 = 𝑋 → if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ) = if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
1514mpteq2dv 5252 . . . 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 2727 . . . 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 7457 . . . . . 6 (ℕ0m 𝐼) ∈ V
182, 17rabex2 5338 . . . . 5 𝐷 ∈ V
1918mptex 7239 . . . 4 (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V
2015, 16, 19fvmpt 7008 . . 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 2767 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 1533  wcel 2098  {crab 3428  ifcif 4530  cmpt 5233  ccnv 5679  cima 5683  cfv 6551  (class class class)co 7424  m cmap 8849  Fincfn 8968  0cc0 11144  1c1 11145  cn 12248  0cn0 12508  0gc0g 17426  1rcur 20126   mVar cmvr 21843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mvr 21848
This theorem is referenced by:  mvrval2  21930  mplcoe3  21981  evlslem1  22033
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