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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 1187 | . . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → 𝐹:𝐴⟶ℝ) | |
2 | suppssdm 7837 | . . . . . . . 8 ⊢ (𝐺 supp 0) ⊆ dom 𝐺 | |
3 | fdm 6516 | . . . . . . . 8 ⊢ (𝐺:𝐴⟶ℝ → dom 𝐺 = 𝐴) | |
4 | 2, 3 | sseqtrid 4018 | . . . . . . 7 ⊢ (𝐺:𝐴⟶ℝ → (𝐺 supp 0) ⊆ 𝐴) |
5 | 4 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐺 supp 0) ⊆ 𝐴) |
6 | 5 | sselda 3966 | . . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → 𝑥 ∈ 𝐴) |
7 | 1, 6 | ffvelrnd 6846 | . . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → (𝐹‘𝑥) ∈ ℝ) |
8 | simpl2 1188 | . . . . 5 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → 𝐺:𝐴⟶ℝ) | |
9 | 8, 6 | ffvelrnd 6846 | . . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → (𝐺‘𝑥) ∈ ℝ) |
10 | ffn 6508 | . . . . . . 7 ⊢ (𝐺:𝐴⟶ℝ → 𝐺 Fn 𝐴) | |
11 | 10 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴) |
12 | simp3 1134 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
13 | 0red 10638 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → 0 ∈ ℝ) | |
14 | elsuppfn 7832 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 0 ∈ ℝ) → (𝑥 ∈ (𝐺 supp 0) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) ≠ 0))) | |
15 | 11, 12, 13, 14 | syl3anc 1367 | . . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐺 supp 0) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) ≠ 0))) |
16 | 15 | simplbda 502 | . . . 4 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → (𝐺‘𝑥) ≠ 0) |
17 | 7, 9, 16 | redivcld 11462 | . . 3 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐺 supp 0)) → ((𝐹‘𝑥) / (𝐺‘𝑥)) ∈ ℝ) |
18 | 17 | fmpttd 6873 | . 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 10588 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → ℝ ⊆ ℂ) |
22 | 19, 21 | fssd 6522 | . . . . 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 6522 | . . . . 5 ⊢ (𝐺:𝐴⟶ℝ → 𝐺:𝐴⟶ℂ) |
26 | id 22 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
27 | 22, 25, 26 | 3anim123i 1147 | . . . 4 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉)) |
28 | fdivmpt 44594 | . . . 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 6493 | . 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 259 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ↦ cmpt 5138 dom cdm 5549 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 supp csupp 7824 ℂcc 10529 ℝcr 10530 0cc0 10531 / cdiv 11291 /f cfdiv 44591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-supp 7825 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-fdiv 44592 |
This theorem is referenced by: refdivpm 44598 elbigolo1 44611 |
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