| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elpclN | Structured version Visualization version GIF version | ||
| Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| pclfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| pclfval.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
| pclfval.c | ⊢ 𝑈 = (PCl‘𝐾) |
| elpcl.q | ⊢ 𝑄 ∈ V |
| Ref | Expression |
|---|---|
| elpclN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pclfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 2 | pclfval.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
| 3 | pclfval.c | . . . 4 ⊢ 𝑈 = (PCl‘𝐾) | |
| 4 | 1, 2, 3 | pclvalN 39892 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
| 5 | 4 | eleq2d 2827 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})) |
| 6 | elpcl.q | . . 3 ⊢ 𝑄 ∈ V | |
| 7 | 6 | elintrab 4960 | . 2 ⊢ (𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦)) |
| 8 | 5, 7 | bitrdi 287 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∩ cint 4946 ‘cfv 6561 Atomscatm 39264 PSubSpcpsubsp 39498 PClcpclN 39889 |
| 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-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-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-psubsp 39505 df-pclN 39890 |
| This theorem is referenced by: pclfinclN 39952 |
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