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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fdivmptfv | Structured version Visualization version GIF version | ||
| Description: The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.) |
| Ref | Expression |
|---|---|
| fdivmptfv | β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β ((πΉ /f πΊ)βπ) = ((πΉβπ) / (πΊβπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fdivmpt 47380 | . . 3 β’ ((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β (πΉ /f πΊ) = (π₯ β (πΊ supp 0) β¦ ((πΉβπ₯) / (πΊβπ₯)))) | |
| 2 | 1 | adantr 480 | . 2 β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β (πΉ /f πΊ) = (π₯ β (πΊ supp 0) β¦ ((πΉβπ₯) / (πΊβπ₯)))) |
| 3 | fveq2 6881 | . . . 4 β’ (π₯ = π β (πΉβπ₯) = (πΉβπ)) | |
| 4 | fveq2 6881 | . . . 4 β’ (π₯ = π β (πΊβπ₯) = (πΊβπ)) | |
| 5 | 3, 4 | oveq12d 7419 | . . 3 β’ (π₯ = π β ((πΉβπ₯) / (πΊβπ₯)) = ((πΉβπ) / (πΊβπ))) |
| 6 | 5 | adantl 481 | . 2 β’ ((((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β§ π₯ = π) β ((πΉβπ₯) / (πΊβπ₯)) = ((πΉβπ) / (πΊβπ))) |
| 7 | simpr 484 | . 2 β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β π β (πΊ supp 0)) | |
| 8 | ovexd 7436 | . 2 β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β ((πΉβπ) / (πΊβπ)) β V) | |
| 9 | 2, 6, 7, 8 | fvmptd 6995 | 1 β’ (((πΉ:π΄βΆβ β§ πΊ:π΄βΆβ β§ π΄ β π) β§ π β (πΊ supp 0)) β ((πΉ /f πΊ)βπ) = ((πΉβπ) / (πΊβπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3466 β¦ cmpt 5221 βΆwf 6529 βcfv 6533 (class class class)co 7401 supp csupp 8140 βcc 11103 0cc0 11105 / cdiv 11867 /f cfdiv 47377 |
| 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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-supp 8141 df-fdiv 47378 |
| This theorem is referenced by: (None) |
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