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Mirrors > Home > MPE Home > Th. List > Mathboxes > poml6N | Structured version Visualization version GIF version |
Description: Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
poml6.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
poml6.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
poml6N | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1193 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝐾 ∈ HL) | |
2 | simpl2 1194 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ 𝐶) | |
3 | eqid 2739 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | poml6.c | . . . . 5 ⊢ 𝐶 = (PSubCl‘𝐾) | |
5 | 3, 4 | psubclssatN 37729 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾)) |
6 | 1, 2, 5 | syl2anc 587 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ (Atoms‘𝐾)) |
7 | simpl3 1195 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑌 ∈ 𝐶) | |
8 | 3, 4 | psubclssatN 37729 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → 𝑌 ⊆ (Atoms‘𝐾)) |
9 | 1, 7, 8 | syl2anc 587 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑌 ⊆ (Atoms‘𝐾)) |
10 | simpr 488 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ 𝑌) | |
11 | poml6.p | . . . . 5 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
12 | 11, 4 | psubcli2N 37727 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
13 | 1, 7, 12 | syl2anc 587 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
14 | 3, 11 | poml4N 37741 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))) |
15 | 14 | imp 410 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ (𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
16 | 1, 6, 9, 10, 13, 15 | syl32anc 1380 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
17 | 11, 4 | psubcli2N 37727 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
18 | 1, 2, 17 | syl2anc 587 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
19 | 16, 18 | eqtrd 2779 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∩ cin 3882 ⊆ wss 3883 ‘cfv 6401 Atomscatm 37051 HLchlt 37138 ⊥𝑃cpolN 37690 PSubClcpscN 37722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5196 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 ax-riotaBAD 36741 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-riota 7192 df-ov 7238 df-oprab 7239 df-undef 8039 df-proset 17835 df-poset 17853 df-plt 17869 df-lub 17885 df-glb 17886 df-join 17887 df-meet 17888 df-p0 17964 df-p1 17965 df-lat 17971 df-clat 18038 df-oposet 36964 df-ol 36966 df-oml 36967 df-covers 37054 df-ats 37055 df-atl 37086 df-cvlat 37110 df-hlat 37139 df-pmap 37292 df-polarityN 37691 df-psubclN 37723 |
This theorem is referenced by: osumcllem9N 37752 |
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