Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > refdivmptfv | Structured version Visualization version GIF version |
Description: The function value of a quotient of two functions into the real numbers. (Contributed by AV, 19-May-2020.) |
Ref | Expression |
---|---|
refdivmptfv | ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℝ) | |
2 | ax-resscn 10594 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐹:𝐴⟶ℝ → ℝ ⊆ ℂ) |
4 | 1, 3 | fssd 6528 | . . . . 5 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ) |
5 | id 22 | . . . . . 6 ⊢ (𝐺:𝐴⟶ℝ → 𝐺:𝐴⟶ℝ) | |
6 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐺:𝐴⟶ℝ → ℝ ⊆ ℂ) |
7 | 5, 6 | fssd 6528 | . . . . 5 ⊢ (𝐺:𝐴⟶ℝ → 𝐺:𝐴⟶ℂ) |
8 | id 22 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
9 | 4, 7, 8 | 3anim123i 1147 | . . . 4 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉)) |
10 | fdivmpt 44620 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) |
12 | 11 | adantr 483 | . 2 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))) |
13 | fveq2 6670 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
14 | fveq2 6670 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
15 | 13, 14 | oveq12d 7174 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) / (𝐺‘𝑥)) = ((𝐹‘𝑋) / (𝐺‘𝑋))) |
16 | 15 | adantl 484 | . 2 ⊢ ((((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥) / (𝐺‘𝑥)) = ((𝐹‘𝑋) / (𝐺‘𝑋))) |
17 | simpr 487 | . 2 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → 𝑋 ∈ (𝐺 supp 0)) | |
18 | ovexd 7191 | . 2 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹‘𝑋) / (𝐺‘𝑋)) ∈ V) | |
19 | 12, 16, 17, 18 | fvmptd 6775 | 1 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supp csupp 7830 ℂcc 10535 ℝcr 10536 0cc0 10537 / cdiv 11297 /f cfdiv 44617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-supp 7831 df-fdiv 44618 |
This theorem is referenced by: elbigolo1 44637 |
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