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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 8139. (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 9266 | . . . . . 6 ⊢ Rel finSupp | |
| 3 | 2 | brrelex1i 5680 | . . . . 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 7060 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴):𝐷⟶𝑉) |
| 7 | dmfex 7847 | . . . 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 7170 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) ∈ V) |
| 10 | fsuppssov1.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
| 11 | funmpt 6530 | . . 3 ⊢ Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) | |
| 12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵))) |
| 13 | ssidd 3957 | . . 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 5681 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌 → 𝑌 ∈ V) |
| 17 | 1, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
| 18 | 13, 14, 5, 15, 17 | suppssov1 8139 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌)) |
| 19 | 9, 10, 12, 1, 18 | fsuppsssuppgd 9285 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 class class class wbr 5098 ↦ cmpt 5179 Fun wfun 6486 ⟶wf 6488 (class class class)co 7358 supp csupp 8102 finSupp cfsupp 9264 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-supp 8103 df-1o 8397 df-en 8884 df-fin 8887 df-fsupp 9265 |
| This theorem is referenced by: elrgspn 33328 elrgspnsubrunlem2 33330 evlextv 33707 mplvrpmrhm 33712 selvvvval 42824 evlselv 42826 |
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