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Mirrors > Home > MPE Home > Th. List > suppssof1 | Structured version Visualization version GIF version |
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
suppssof1.s | ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿) |
suppssof1.o | ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) |
suppssof1.a | ⊢ (𝜑 → 𝐴:𝐷⟶𝑉) |
suppssof1.b | ⊢ (𝜑 → 𝐵:𝐷⟶𝑅) |
suppssof1.d | ⊢ (𝜑 → 𝐷 ∈ 𝑊) |
suppssof1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
Ref | Expression |
---|---|
suppssof1 | ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssof1.a | . . . . 5 ⊢ (𝜑 → 𝐴:𝐷⟶𝑉) | |
2 | 1 | ffnd 6712 | . . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐷) |
3 | suppssof1.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅) | |
4 | 3 | ffnd 6712 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐷) |
5 | suppssof1.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
6 | inidm 4213 | . . . 4 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
7 | eqidd 2727 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥)) | |
8 | eqidd 2727 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥)) | |
9 | 2, 4, 5, 5, 6, 7, 8 | offval 7676 | . . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
10 | 9 | oveq1d 7420 | . 2 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) = ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍)) |
11 | 1 | feqmptd 6954 | . . . . 5 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
12 | 11 | oveq1d 7420 | . . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) = ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌)) |
13 | suppssof1.s | . . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿) | |
14 | 12, 13 | eqsstrrd 4016 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌) ⊆ 𝐿) |
15 | suppssof1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
16 | fvexd 6900 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) | |
17 | 3 | ffvelcdmda 7080 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅) |
18 | suppssof1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
19 | 14, 15, 16, 17, 18 | suppssov1 8183 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍) ⊆ 𝐿) |
20 | 10, 19 | eqsstrd 4015 | 1 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 ↦ cmpt 5224 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∘f cof 7665 supp csupp 8146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-supp 8147 |
This theorem is referenced by: frlmup1 21693 psrbagev1 21980 psrbagev1OLD 21981 jensen 26876 offinsupp1 32459 |
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