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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fsuppssindlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for fsuppssind 42603. 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 6977 | . 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 3961 | . . . . . 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 8220 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 ) |
| 10 | 9 | eqcomd 2743 | . . . . . 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 4562 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥)) |
| 14 | 13 | mpteq2dva 5242 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) |
| 15 | 2, 14 | eqtr4d 2780 | 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 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 ifcif 4525 ↦ cmpt 5225 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8186 |
| This theorem is referenced by: fsuppssind 42603 |
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