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Theorem fdivmpt 47313
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 7229 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 ∈ V)
213adant2 1129 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 ∈ V)
3 fex 7229 . . . 4 ((𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ V)
433adant1 1128 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ V)
5 fdivval 47312 . . . 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 2770 . . 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 582 . 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 6716 . . . . 5 (𝐹:π΄βŸΆβ„‚ β†’ 𝐹 Fn 𝐴)
1093ad2ant1 1131 . . . 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 6725 . . . . . . 7 (𝐺:π΄βŸΆβ„‚ β†’ dom 𝐺 = 𝐴)
1312eqcomd 2736 . . . . . 6 (𝐺:π΄βŸΆβ„‚ β†’ 𝐴 = dom 𝐺)
14133ad2ant2 1132 . . . . 5 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐴 = dom 𝐺)
1511, 14sseqtrrid 4034 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 supp 0) βŠ† 𝐴)
16 fnssres 6672 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐺 supp 0) βŠ† 𝐴) β†’ (𝐹 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
1710, 15, 16syl2anc 582 . . 3 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
18 ffn 6716 . . . . 5 (𝐺:π΄βŸΆβ„‚ β†’ 𝐺 Fn 𝐴)
19183ad2ant2 1132 . . . 4 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐺:π΄βŸΆβ„‚ ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 Fn 𝐴)
20 fnssres 6672 . . . 4 ((𝐺 Fn 𝐴 ∧ (𝐺 supp 0) βŠ† 𝐴) β†’ (𝐺 β†Ύ (𝐺 supp 0)) Fn (𝐺 supp 0))
2119, 15, 20syl2anc 582 . . 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 4217 . . 3 ((𝐺 supp 0) ∩ (𝐺 supp 0)) = (𝐺 supp 0)
24 fvres 6909 . . . 4 (π‘₯ ∈ (𝐺 supp 0) β†’ ((𝐹 β†Ύ (𝐺 supp 0))β€˜π‘₯) = (πΉβ€˜π‘₯))
2524adantl 480 . . 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 6909 . . . 4 (π‘₯ ∈ (𝐺 supp 0) β†’ ((𝐺 β†Ύ (𝐺 supp 0))β€˜π‘₯) = (πΊβ€˜π‘₯))
2726adantl 480 . . 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 2770 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3947   ↦ cmpt 5230  dom cdm 5675   β†Ύ cres 5677   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∘f cof 7670   supp csupp 8148  β„‚cc 11110  0cc0 11112   / cdiv 11875   /f cfdiv 47310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-supp 8149  df-fdiv 47311
This theorem is referenced by:  fdivmptf  47314  refdivmptf  47315  fdivmptfv  47318  refdivmptfv  47319
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