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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsuppssindlem1 | Structured version Visualization version GIF version |
Description: Lemma for fsuppssind 39772. 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 6714 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
3 | fvres 6670 | . . . . 5 ⊢ (𝑥 ∈ 𝑆 → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥)) | |
4 | 3 | adantl 486 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝑆) → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥)) |
5 | eldif 3864 | . . . . . 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 7863 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 ) |
10 | 9 | eqcomd 2765 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → 0 = (𝐹‘𝑥)) |
11 | 5, 10 | sylan2br 598 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝑆)) → 0 = (𝐹‘𝑥)) |
12 | 11 | anassrs 472 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝑆) → 0 = (𝐹‘𝑥)) |
13 | 4, 12 | ifeqda 4449 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥)) |
14 | 13 | mpteq2dva 5120 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
15 | 2, 14 | eqtr4d 2797 | 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 400 = wceq 1539 ∈ wcel 2112 ∖ cdif 3851 ⊆ wss 3854 ifcif 4413 ↦ cmpt 5105 ↾ cres 5519 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 supp csupp 7828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-supp 7829 |
This theorem is referenced by: fsuppssind 39772 |
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