![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsuppssindlem1 | Structured version Visualization version GIF version |
Description: Lemma for fsuppssind 42579. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.) |
Ref | Expression |
---|---|
fsuppssindlem1.z | ⊢ (𝜑 → 0 ∈ 𝑊) |
fsuppssindlem1.v | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
fsuppssindlem1.1 | ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
fsuppssindlem1.2 | ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆) |
Ref | Expression |
---|---|
fsuppssindlem1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppssindlem1.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
2 | 1 | feqmptd 6976 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
3 | fvres 6925 | . . . . 5 ⊢ (𝑥 ∈ 𝑆 → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥)) | |
4 | 3 | adantl 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝑆) → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥)) |
5 | eldif 3972 | . . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝑆) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝑆)) | |
6 | fsuppssindlem1.2 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆) | |
7 | fsuppssindlem1.v | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
8 | fsuppssindlem1.z | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝑊) | |
9 | 1, 6, 7, 8 | suppssr 8218 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 ) |
10 | 9 | eqcomd 2740 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → 0 = (𝐹‘𝑥)) |
11 | 5, 10 | sylan2br 595 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝑆)) → 0 = (𝐹‘𝑥)) |
12 | 11 | anassrs 467 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝑆) → 0 = (𝐹‘𝑥)) |
13 | 4, 12 | ifeqda 4566 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥)) |
14 | 13 | mpteq2dva 5247 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
15 | 2, 14 | eqtr4d 2777 | 1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ⊆ wss 3962 ifcif 4530 ↦ cmpt 5230 ↾ cres 5690 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-supp 8184 |
This theorem is referenced by: fsuppssind 42579 |
Copyright terms: Public domain | W3C validator |