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Mirrors > Home > MPE Home > Th. List > Mathboxes > fisuppov1 | Structured version Visualization version GIF version |
Description: Formula building theorem for finite support: operator with left annihilator. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
Ref | Expression |
---|---|
fisuppov1.1 | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
fisuppov1.2 | ⊢ (𝜑 → 0 ∈ 𝑋) |
fisuppov1.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
fisuppov1.4 | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
fisuppov1.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌) |
fisuppov1.6 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐸) |
fisuppov1.7 | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
fisuppov1.8 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍) |
Ref | Expression |
---|---|
fisuppov1 | ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fisuppov1.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
2 | fisuppov1.4 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
3 | 1, 2 | ssexd 5329 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
4 | 3 | mptexd 7243 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V) |
5 | fisuppov1.1 | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
6 | funmpt 6605 | . . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵))) |
8 | fisuppov1.7 | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
9 | fisuppov1.6 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐸) | |
10 | 9, 2 | feqresmpt 6977 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
11 | 10 | oveq1d 7445 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 )) |
12 | 9, 1 | fexd 7246 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | fisuppov1.2 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋) | |
14 | ressuppss 8206 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ 𝑋) → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 )) | |
15 | 12, 13, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) ⊆ (𝐹 supp 0 )) |
16 | 11, 15 | eqsstrrd 4034 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 ) ⊆ (𝐹 supp 0 )) |
17 | fisuppov1.8 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍) | |
18 | fvexd 6921 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ V) | |
19 | fisuppov1.5 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌) | |
20 | 16, 17, 18, 19, 13 | suppssov1 8220 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 )) |
21 | 4, 5, 7, 8, 20 | fsuppsssuppgd 9419 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 class class class wbr 5147 ↦ cmpt 5230 ↾ cres 5690 Fun wfun 6556 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 finSupp cfsupp 9398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-supp 8184 df-1o 8504 df-en 8984 df-fin 8987 df-fsupp 9399 |
This theorem is referenced by: elrgspnlem1 33231 elrgspnlem2 33232 |
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