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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 6736 | . . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐷) | 
| 3 | suppssof1.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅) | |
| 4 | 3 | ffnd 6736 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐷) | 
| 5 | suppssof1.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
| 6 | inidm 4226 | . . . 4 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
| 7 | eqidd 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥)) | |
| 8 | eqidd 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥)) | |
| 9 | 2, 4, 5, 5, 6, 7, 8 | offval 7707 | . . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) | 
| 10 | 9 | oveq1d 7447 | . 2 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) = ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍)) | 
| 11 | 1 | feqmptd 6976 | . . . . 5 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) | 
| 12 | 11 | oveq1d 7447 | . . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) = ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌)) | 
| 13 | suppssof1.s | . . . 4 ⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿) | |
| 14 | 12, 13 | eqsstrrd 4018 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) supp 𝑌) ⊆ 𝐿) | 
| 15 | suppssof1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
| 16 | fvexd 6920 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) | |
| 17 | 3 | ffvelcdmda 7103 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅) | 
| 18 | suppssof1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
| 19 | 14, 15, 16, 17, 18 | suppssov1 8223 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) supp 𝑍) ⊆ 𝐿) | 
| 20 | 10, 19 | eqsstrd 4017 | 1 ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 supp csupp 8186 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-supp 8187 | 
| This theorem is referenced by: frlmup1 21819 psrbagev1 22102 jensen 27033 offinsupp1 32739 | 
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