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Theorem fdivmpt 47314
Description: The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.)
Assertion
Ref Expression
fdivmpt ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 /f 𝐺) = (π‘₯ ∈ (𝐺 supp 0) ↦ ((πΉβ€˜π‘₯) / (πΊβ€˜π‘₯))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐹   π‘₯,𝐺   π‘₯,𝑉
Proof of Theorem fdivmpt
StepHypRef Expression
1 fex 7230 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 ∈ V)
213adant2 1131 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 ∈ V)
3 fex 7230 . . . 4 ((𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ V)
433adant1 1130 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ V)
5 fdivval 47313 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) β†’ (𝐹 /f 𝐺) = ((𝐹 ∘f / 𝐺) β†Ύ (𝐺 supp 0)))
6 offres 7972 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) β†’ ((𝐹 ∘f / 𝐺) β†Ύ (𝐺 supp 0)) = ((𝐹 β†Ύ (𝐺 supp 0)) ∘f / (𝐺 β†Ύ (𝐺 supp 0))))
75, 6eqtrd 2772 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V) β†’ (𝐹 /f 𝐺) = ((𝐹 β†Ύ (𝐺 supp 0)) ∘f / (𝐺 β†Ύ (𝐺 supp 0))))
82, 4, 7syl2anc 584 . 2 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 /f 𝐺) = ((𝐹 β†Ύ (𝐺 supp 0)) ∘f / (𝐺 β†Ύ (𝐺 supp 0))))
9 ffn 6717 . . . . 5 (𝐹:π΄βŸΆβ„‚ β†’ 𝐹 Fn 𝐴)
1093ad2ant1 1133 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 Fn 𝐴)
11 suppssdm 8164 . . . . 5 (𝐺 supp 0) βŠ† dom 𝐺
12 fdm 6726 . . . . . . 7 (𝐺:π΄βŸΆβ„‚ β†’ dom 𝐺 = 𝐴)
1312eqcomd 2738 . . . . . 6 (𝐺:π΄βŸΆβ„‚ β†’ 𝐴 = dom 𝐺)
14133ad2ant2 1134 . . . . 5 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐴 = dom 𝐺)
1511, 14sseqtrrid 4035 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 supp 0) βŠ† 𝐴)
16 fnssres 6673 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐺 supp 0) βŠ† 𝐴) β†’ (𝐹 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
1710, 15, 16syl2anc 584 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
18 ffn 6717 . . . . 5 (𝐺:π΄βŸΆβ„‚ β†’ 𝐺 Fn 𝐴)
19183ad2ant2 1134 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 Fn 𝐴)
20 fnssres 6673 . . . 4 ((𝐺 Fn 𝐴 ∧ (𝐺 supp 0) βŠ† 𝐴) β†’ (𝐺 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
2119, 15, 20syl2anc 584 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
22 ovexd 7446 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 supp 0) ∈ V)
23 inidm 4218 . . 3 ((𝐺 supp 0) ∩ (𝐺 supp 0)) = (𝐺 supp 0)
24 fvres 6910 . . . 4 (π‘₯ ∈ (𝐺 supp 0) β†’ ((𝐹 β†Ύ (𝐺 supp 0))β€˜π‘₯) = (πΉβ€˜π‘₯))
2524adantl 482 . . 3 (((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ (𝐺 supp 0)) β†’ ((𝐹 β†Ύ (𝐺 supp 0))β€˜π‘₯) = (πΉβ€˜π‘₯))
26 fvres 6910 . . . 4 (π‘₯ ∈ (𝐺 supp 0) β†’ ((𝐺 β†Ύ (𝐺 supp 0))β€˜π‘₯) = (πΊβ€˜π‘₯))
2726adantl 482 . . 3 (((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ (𝐺 supp 0)) β†’ ((𝐺 β†Ύ (𝐺 supp 0))β€˜π‘₯) = (πΊβ€˜π‘₯))
2817, 21, 22, 22, 23, 25, 27offval 7681 . 2 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ ((𝐹 β†Ύ (𝐺 supp 0)) ∘f / (𝐺 β†Ύ (𝐺 supp 0))) = (π‘₯ ∈ (𝐺 supp 0) ↦ ((πΉβ€˜π‘₯) / (πΊβ€˜π‘₯))))
298, 28eqtrd 2772 1 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 /f 𝐺) = (π‘₯ ∈ (𝐺 supp 0) ↦ ((πΉβ€˜π‘₯) / (πΊβ€˜π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948   ↦ cmpt 5231  dom cdm 5676   β†Ύ cres 5678   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∘f cof 7670   supp csupp 8148  β„‚cc 11110  0cc0 11112   / cdiv 11875   /f cfdiv 47311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-supp 8149  df-fdiv 47312
This theorem is referenced by:  fdivmptf  47315  refdivmptf  47316  fdivmptfv  47319  refdivmptfv  47320
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