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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 8121. (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 9241 | . . . . . 6 ⊢ Rel finSupp | |
| 3 | 2 | brrelex1i 5669 | . . . . 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 7042 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉) |
| 7 | dmfex 7829 | . . . 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 7152 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V) |
| 10 | fsuppssov1.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
| 11 | funmpt 6514 | . . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) | |
| 12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))) |
| 13 | ssidd 3955 | . . 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 5670 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → 𝑌 ∈ V) |
| 17 | 1, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
| 18 | 13, 14, 5, 15, 17 | suppssov1 8121 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌)) |
| 19 | 9, 10, 12, 1, 18 | fsuppsssuppgd 9260 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3433 class class class wbr 5088 ↦ cmpt 5169 Fun wfun 6470 ⟶wf 6472 (class class class)co 7340 supp csupp 8084 finSupp cfsupp 9239 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pr 5367 ax-un 7662 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7343 df-oprab 7344 df-mpo 7345 df-om 7791 df-supp 8085 df-1o 8379 df-en 8864 df-fin 8867 df-fsupp 9240 |
| This theorem is referenced by: elrgspn 33181 elrgspnsubrunlem2 33183 mplvrpmrhm 33545 selvvvval 42575 evlselv 42577 |
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