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Mirrors > Home > MPE Home > Th. List > Mathboxes > refdivmptf | Structured version Visualization version GIF version |
Description: The quotient of two functions into the real numbers is a function into the real numbers. (Contributed by AV, 16-May-2020.) |
Ref | Expression |
---|---|
refdivmptf | ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1199 | . . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → 𝐹:𝐴⟶ℝ) | |
2 | suppssdm 7589 | . . . . . . . 8 ⊢ (𝐺 supp 0) ⊆ dom 𝐺 | |
3 | fdm 6299 | . . . . . . . 8 ⊢ (𝐺:𝐴⟶ℝ → dom 𝐺 = 𝐴) | |
4 | 2, 3 | syl5sseq 3871 | . . . . . . 7 ⊢ (𝐺:𝐴⟶ℝ → (𝐺 supp 0) ⊆ 𝐴) |
5 | 4 | 3ad2ant2 1125 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐺 supp 0) ⊆ 𝐴) |
6 | 5 | sselda 3820 | . . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → 𝑥 ∈ 𝐴) |
7 | 1, 6 | ffvelrnd 6624 | . . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → (𝐹‘𝑥) ∈ ℝ) |
8 | simpl2 1201 | . . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → 𝐺:𝐴⟶ℝ) | |
9 | 8, 6 | ffvelrnd 6624 | . . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → (𝐺‘𝑥) ∈ ℝ) |
10 | ffn 6291 | . . . . . . 7 ⊢ (𝐺:𝐴⟶ℝ → 𝐺 Fn 𝐴) | |
11 | 10 | 3ad2ant2 1125 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴) |
12 | simp3 1129 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
13 | 0red 10380 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → 0 ∈ ℝ) | |
14 | elsuppfn 7584 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 0 ∈ ℝ) → (𝑥 ∈ (𝐺 supp 0) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) ≠ 0))) | |
15 | 11, 12, 13, 14 | syl3anc 1439 | . . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐺 supp 0) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) ≠ 0))) |
16 | 15 | simplbda 495 | . . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → (𝐺‘𝑥) ≠ 0) |
17 | 7, 9, 16 | redivcld 11203 | . . 3 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → ((𝐹‘𝑥) / (𝐺‘𝑥)) ∈ ℝ) |
18 | 17 | fmpttd 6649 | . 2 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))):(𝐺 supp 0)⟶ℝ) |
19 | id 22 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℝ) | |
20 | ax-resscn 10329 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → ℝ ⊆ ℂ) |
22 | 19, 21 | fssd 6305 | . . . . 5 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ) |
23 | id 22 | . . . . . 6 ⊢ (𝐺:𝐴⟶ℝ → 𝐺:𝐴⟶ℝ) | |
24 | 20 | a1i 11 | . . . . . 6 ⊢ (𝐺:𝐴⟶ℝ → ℝ ⊆ ℂ) |
25 | 23, 24 | fssd 6305 | . . . . 5 ⊢ (𝐺:𝐴⟶ℝ → 𝐺:𝐴⟶ℂ) |
26 | id 22 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
27 | 22, 25, 26 | 3anim123i 1151 | . . . 4 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉)) |
28 | fdivmpt 43342 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) | |
29 | 27, 28 | syl 17 | . . 3 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) |
30 | 29 | feq1d 6276 | . 2 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → ((𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ ↔ (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))):(𝐺 supp 0)⟶ℝ)) |
31 | 18, 30 | mpbird 249 | 1 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ⊆ wss 3791 ↦ cmpt 4965 dom cdm 5355 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 supp csupp 7576 ℂcc 10270 ℝcr 10271 0cc0 10272 / cdiv 11032 /f cfdiv 43339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-supp 7577 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-fdiv 43340 |
This theorem is referenced by: refdivpm 43346 elbigolo1 43359 |
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