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Mirrors > Home > MPE Home > Th. List > fsuppssov1 | Structured version Visualization version GIF version |
Description: Formula building theorem for finite support: operator with left annihilator. Finite support version of suppssov1 8221. (Contributed by SN, 26-Apr-2025.) |
Ref | Expression |
---|---|
fsuppssov1.s | ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌) |
fsuppssov1.o | ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) |
fsuppssov1.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉) |
fsuppssov1.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅) |
fsuppssov1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
Ref | Expression |
---|---|
fsuppssov1 | ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppssov1.s | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌) | |
2 | relfsupp 9401 | . . . . . 6 ⊢ Rel finSupp | |
3 | 2 | brrelex1i 5745 | . . . . 5 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → (𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V) |
5 | fsuppssov1.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉) | |
6 | 5 | fmpttd 7135 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉) |
7 | dmfex 7928 | . . . 4 ⊢ (((𝑥 ∈ 𝐷 ↦ 𝐴) ∈ V ∧ (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉) → 𝐷 ∈ V) | |
8 | 4, 6, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
9 | 8 | mptexd 7244 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V) |
10 | fsuppssov1.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
11 | funmpt 6606 | . . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))) |
13 | ssidd 4019 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌)) | |
14 | fsuppssov1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
15 | fsuppssov1.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅) | |
16 | 2 | brrelex2i 5746 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → 𝑌 ∈ V) |
17 | 1, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
18 | 13, 14, 5, 15, 17 | suppssov1 8221 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌)) |
19 | 9, 10, 12, 1, 18 | fsuppsssuppgd 9420 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 ↦ cmpt 5231 Fun wfun 6557 ⟶wf 6559 (class class class)co 7431 supp csupp 8184 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: selvvvval 42572 evlselv 42574 |
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