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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdivpm | Structured version Visualization version GIF version |
Description: The quotient of two functions into the complex numbers is a partial function. (Contributed by AV, 16-May-2020.) |
Ref | Expression |
---|---|
fdivpm | ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10607 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → ℂ ∈ V) |
3 | simp3 1135 | . 2 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
4 | fdivmptf 44955 | . 2 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ) | |
5 | suppssdm 7826 | . . 3 ⊢ (𝐺 supp 0) ⊆ dom 𝐺 | |
6 | fdm 6495 | . . . . 5 ⊢ (𝐺:𝐴⟶ℂ → dom 𝐺 = 𝐴) | |
7 | 6 | eqcomd 2804 | . . . 4 ⊢ (𝐺:𝐴⟶ℂ → 𝐴 = dom 𝐺) |
8 | 7 | 3ad2ant2 1131 | . . 3 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → 𝐴 = dom 𝐺) |
9 | 5, 8 | sseqtrrid 3968 | . 2 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐺 supp 0) ⊆ 𝐴) |
10 | elpm2r 8407 | . 2 ⊢ (((ℂ ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ((𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ ∧ (𝐺 supp 0) ⊆ 𝐴)) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴)) | |
11 | 2, 3, 4, 9, 10 | syl22anc 837 | 1 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 dom cdm 5519 ⟶wf 6320 (class class class)co 7135 supp csupp 7813 ↑pm cpm 8390 ℂcc 10524 0cc0 10526 /f cfdiv 44951 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-supp 7814 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-fdiv 44952 |
This theorem is referenced by: (None) |
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