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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 35176 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) |
5 | 4 | eleq2d 2687 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})) |
6 | elpcl.q | . . 3 ⊢ 𝑄 ∈ V | |
7 | 6 | elintrab 4488 | . 2 ⊢ (𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦)) |
8 | 5, 7 | syl6bb 276 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∩ cint 4475 ‘cfv 5888 Atomscatm 34550 PSubSpcpsubsp 34782 PClcpclN 35173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-psubsp 34789 df-pclN 35174 |
This theorem is referenced by: pclfinclN 35236 |
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