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Theorem fdivval 42098
Description: The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
Assertion
Ref Expression
fdivval ((𝐹𝑉𝐺𝑊) → (𝐹 /f 𝐺) = ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)))

Proof of Theorem fdivval
Dummy variables 𝑥 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fdiv 42097 . . 3 /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓𝑓 / 𝑔) ↾ (𝑔 supp 0)))
21a1i 11 . 2 ((𝐹𝑉𝐺𝑊) → /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓𝑓 / 𝑔) ↾ (𝑔 supp 0))))
3 oveq12 6644 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑓 / 𝑔) = (𝐹𝑓 / 𝐺))
4 oveq1 6642 . . . . 5 (𝑔 = 𝐺 → (𝑔 supp 0) = (𝐺 supp 0))
54adantl 482 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔 supp 0) = (𝐺 supp 0))
63, 5reseq12d 5386 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑓 / 𝑔) ↾ (𝑔 supp 0)) = ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)))
76adantl 482 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → ((𝑓𝑓 / 𝑔) ↾ (𝑔 supp 0)) = ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)))
8 elex 3207 . . 3 (𝐹𝑉𝐹 ∈ V)
98adantr 481 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
10 elex 3207 . . 3 (𝐺𝑊𝐺 ∈ V)
1110adantl 482 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
12 funmpt 5914 . . . 4 Fun (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) / (𝐺𝑥)))
13 offval0 42064 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹𝑓 / 𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) / (𝐺𝑥))))
1413funeqd 5898 . . . 4 ((𝐹𝑉𝐺𝑊) → (Fun (𝐹𝑓 / 𝐺) ↔ Fun (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) / (𝐺𝑥)))))
1512, 14mpbiri 248 . . 3 ((𝐹𝑉𝐺𝑊) → Fun (𝐹𝑓 / 𝐺))
16 ovex 6663 . . 3 (𝐺 supp 0) ∈ V
17 resfunexg 6464 . . 3 ((Fun (𝐹𝑓 / 𝐺) ∧ (𝐺 supp 0) ∈ V) → ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)) ∈ V)
1815, 16, 17sylancl 693 . 2 ((𝐹𝑉𝐺𝑊) → ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)) ∈ V)
192, 7, 9, 11, 18ovmpt2d 6773 1 ((𝐹𝑉𝐺𝑊) → (𝐹 /f 𝐺) = ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  cmpt 4720  dom cdm 5104  cres 5106  Fun wfun 5870  cfv 5876  (class class class)co 6635  cmpt2 6637  𝑓 cof 6880   supp csupp 7280  0cc0 9921   / cdiv 10669   /f cfdiv 42096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-fdiv 42097
This theorem is referenced by:  fdivmpt  42099
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