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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 5324 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
| 4 | 3 | mptexd 7244 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) ∈ V) |
| 5 | fisuppov1.1 | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
| 6 | funmpt 6604 | . . 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 6978 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥))) |
| 11 | 10 | oveq1d 7446 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐷) supp 0 ) = ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝑥)) supp 0 )) |
| 12 | 9, 1 | fexd 7247 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
| 13 | fisuppov1.2 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑋) | |
| 14 | ressuppss 8208 | . . . . 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 4019 | . . 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 8222 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) supp 𝑍) ⊆ (𝐹 supp 0 )) |
| 21 | 4, 5, 7, 8, 20 | fsuppsssuppgd 9422 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 finSupp cfsupp 9401 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 |
| This theorem is referenced by: elrgspnlem1 33246 elrgspnlem2 33247 fldextrspunlsplem 33723 |
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