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 3992 | . 2 ⊢ (𝐾 ∈ HL → 𝐴 ⊆ 𝐴) | |
2 | 1psubcl.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | eqid 2823 | . . . . 5 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
4 | 2, 3 | pol1N 37048 | . . . 4 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘𝐴) = ∅) |
5 | 4 | fveq2d 6676 | . . 3 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝐴)) = ((⊥𝑃‘𝐾)‘∅)) |
6 | 2, 3 | pol0N 37047 | . . 3 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘∅) = 𝐴) |
7 | 5, 6 | eqtrd 2858 | . 2 ⊢ (𝐾 ∈ HL → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝐴)) = 𝐴) |
8 | 1psubcl.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
9 | 2, 3, 8 | ispsubclN 37075 | . 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 1537 ∈ wcel 2114 ⊆ wss 3938 ∅c0 4293 ‘cfv 6357 Atomscatm 36401 HLchlt 36488 ⊥𝑃cpolN 37040 PSubClcpscN 37072 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-undef 7941 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-pmap 36642 df-polarityN 37041 df-psubclN 37073 |
This theorem is referenced by: (None) |
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