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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1psubclN | Structured version Visualization version GIF version |
Description: The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1psubcl.a | ⊢ 𝐴 = (Atoms‘𝐾) |
1psubcl.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
1psubclN | ⊢ (𝐾 ∈ HL → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 4005 | . 2 ⊢ (𝐾 ∈ HL → 𝐴 ⊆ 𝐴) | |
2 | 1psubcl.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | eqid 2728 | . . . . 5 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
4 | 2, 3 | pol1N 39415 | . . . 4 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘𝐴) = ∅) |
5 | 4 | fveq2d 6906 | . . 3 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝐴)) = ((⊥𝑃‘𝐾)‘∅)) |
6 | 2, 3 | pol0N 39414 | . . 3 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘∅) = 𝐴) |
7 | 5, 6 | eqtrd 2768 | . 2 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝐴)) = 𝐴) |
8 | 1psubcl.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
9 | 2, 3, 8 | ispsubclN 39442 | . 2 ⊢ (𝐾 ∈ HL → (𝐴 ∈ 𝐶 ↔ (𝐴 ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝐴)) = 𝐴))) |
10 | 1, 7, 9 | mpbir2and 711 | 1 ⊢ (𝐾 ∈ HL → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 ∅c0 4326 ‘cfv 6553 Atomscatm 38767 HLchlt 38854 ⊥𝑃cpolN 39407 PSubClcpscN 39439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-pmap 39009 df-polarityN 39408 df-psubclN 39440 |
This theorem is referenced by: (None) |
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