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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 6517 | . . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐷) |
3 | suppssof1.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅) | |
4 | 3 | ffnd 6517 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐷) |
5 | suppssof1.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
6 | inidm 4197 | . . . 4 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
7 | eqidd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥)) | |
8 | eqidd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥)) | |
9 | 2, 4, 5, 5, 6, 7, 8 | offval 7418 | . . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
10 | 9 | oveq1d 7173 | . 2 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) = ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍)) |
11 | 1 | feqmptd 6735 | . . . . 5 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
12 | 11 | oveq1d 7173 | . . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) = ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌)) |
13 | suppssof1.s | . . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿) | |
14 | 12, 13 | eqsstrrd 4008 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌) ⊆ 𝐿) |
15 | suppssof1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
16 | fvexd 6687 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) | |
17 | 3 | ffvelrnda 6853 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅) |
18 | suppssof1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
19 | 14, 15, 16, 17, 18 | suppssov1 7864 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍) ⊆ 𝐿) |
20 | 10, 19 | eqsstrd 4007 | 1 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 supp csupp 7832 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-supp 7833 |
This theorem is referenced by: psrbagev1 20292 frlmup1 20944 jensen 25568 offinsupp1 30465 |
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