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Mirrors > Home > ILE Home > Th. List > suppssof1 | GIF version |
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
suppssof1.s | ⊢ (𝜑 → (◡𝐴 “ (V ∖ {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) |
Ref | Expression |
---|---|
suppssof1 | ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssof1.a | . . . . . 6 ⊢ (𝜑 → 𝐴:𝐷⟶𝑉) | |
2 | ffn 5195 | . . . . . 6 ⊢ (𝐴:𝐷⟶𝑉 → 𝐴 Fn 𝐷) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐴 Fn 𝐷) |
4 | suppssof1.b | . . . . . 6 ⊢ (𝜑 → 𝐵:𝐷⟶𝑅) | |
5 | ffn 5195 | . . . . . 6 ⊢ (𝐵:𝐷⟶𝑅 → 𝐵 Fn 𝐷) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐵 Fn 𝐷) |
7 | suppssof1.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
8 | inidm 3224 | . . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
9 | eqidd 2096 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥)) | |
10 | eqidd 2096 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥)) | |
11 | 3, 6, 7, 7, 8, 9, 10 | offval 5901 | . . . 4 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
12 | 11 | cnveqd 4643 | . . 3 ⊢ (𝜑 → ◡(𝐴 ∘𝑓 𝑂𝐵) = ◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
13 | 12 | imaeq1d 4806 | . 2 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍}))) |
14 | 1 | feqmptd 5392 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
15 | 14 | cnveqd 4643 | . . . . 5 ⊢ (𝜑 → ◡𝐴 = ◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
16 | 15 | imaeq1d 4806 | . . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) = (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌}))) |
17 | suppssof1.s | . . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿) | |
18 | 16, 17 | eqsstr3d 3076 | . . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿) |
19 | suppssof1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
20 | funfvex 5357 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑥 ∈ dom 𝐴) → (𝐴‘𝑥) ∈ V) | |
21 | 20 | funfni 5148 | . . . 4 ⊢ ((𝐴 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) |
22 | 3, 21 | sylan 278 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) |
23 | 4 | ffvelrnda 5473 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅) |
24 | 18, 19, 22, 23 | suppssov1 5891 | . 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿) |
25 | 13, 24 | eqsstrd 3075 | 1 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 Vcvv 2633 ∖ cdif 3010 ⊆ wss 3013 {csn 3466 ↦ cmpt 3921 ◡ccnv 4466 “ cima 4470 Fn wfn 5044 ⟶wf 5045 ‘cfv 5049 (class class class)co 5690 ∘𝑓 cof 5892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-of 5894 |
This theorem is referenced by: (None) |
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