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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdivval | Structured version Visualization version GIF version |
Description: The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.) |
Ref | Expression |
---|---|
fdivval | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 /f 𝐺) = ((𝐹 ∘𝑓 / 𝐺) ↾ (𝐺 supp 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fdiv 42332 | . . 3 ⊢ /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘𝑓 / 𝑔) ↾ (𝑔 supp 0))) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘𝑓 / 𝑔) ↾ (𝑔 supp 0)))) |
3 | oveq12 6659 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∘𝑓 / 𝑔) = (𝐹 ∘𝑓 / 𝐺)) | |
4 | oveq1 6657 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 supp 0) = (𝐺 supp 0)) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔 supp 0) = (𝐺 supp 0)) |
6 | 3, 5 | reseq12d 5397 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓 ∘𝑓 / 𝑔) ↾ (𝑔 supp 0)) = ((𝐹 ∘𝑓 / 𝐺) ↾ (𝐺 supp 0))) |
7 | 6 | adantl 482 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓 ∘𝑓 / 𝑔) ↾ (𝑔 supp 0)) = ((𝐹 ∘𝑓 / 𝐺) ↾ (𝐺 supp 0))) |
8 | elex 3212 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
9 | 8 | adantr 481 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V) |
10 | elex 3212 | . . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) | |
11 | 10 | adantl 482 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V) |
12 | funmpt 5926 | . . . 4 ⊢ Fun (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))) | |
13 | offval0 42299 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 / 𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) | |
14 | 13 | funeqd 5910 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (Fun (𝐹 ∘𝑓 / 𝐺) ↔ Fun (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))))) |
15 | 12, 14 | mpbiri 248 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → Fun (𝐹 ∘𝑓 / 𝐺)) |
16 | ovex 6678 | . . 3 ⊢ (𝐺 supp 0) ∈ V | |
17 | resfunexg 6479 | . . 3 ⊢ ((Fun (𝐹 ∘𝑓 / 𝐺) ∧ (𝐺 supp 0) ∈ V) → ((𝐹 ∘𝑓 / 𝐺) ↾ (𝐺 supp 0)) ∈ V) | |
18 | 15, 16, 17 | sylancl 694 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘𝑓 / 𝐺) ↾ (𝐺 supp 0)) ∈ V) |
19 | 2, 7, 9, 11, 18 | ovmpt2d 6788 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 /f 𝐺) = ((𝐹 ∘𝑓 / 𝐺) ↾ (𝐺 supp 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ↦ cmpt 4729 dom cdm 5114 ↾ cres 5116 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ∘𝑓 cof 6895 supp csupp 7295 0cc0 9936 / cdiv 10684 /f cfdiv 42331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-fdiv 42332 |
This theorem is referenced by: fdivmpt 42334 |
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