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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 1187 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝐾 ∈ HL) | |
2 | simpl2 1188 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ∈ 𝐶) | |
3 | eqid 2821 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | poml6.c | . . . . 5 ⊢ 𝐶 = (PSubCl‘𝐾) | |
5 | 3, 4 | psubclssatN 37092 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾)) |
6 | 1, 2, 5 | syl2anc 586 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ (Atoms‘𝐾)) |
7 | simpl3 1189 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑌 ∈ 𝐶) | |
8 | 3, 4 | psubclssatN 37092 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → 𝑌 ⊆ (Atoms‘𝐾)) |
9 | 1, 7, 8 | syl2anc 586 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑌 ⊆ (Atoms‘𝐾)) |
10 | simpr 487 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ 𝑌) | |
11 | poml6.p | . . . . 5 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
12 | 11, 4 | psubcli2N 37090 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
13 | 1, 7, 12 | syl2anc 586 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
14 | 3, 11 | poml4N 37104 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))) |
15 | 14 | imp 409 | . . 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 1374 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
17 | 11, 4 | psubcli2N 37090 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
18 | 1, 2, 17 | syl2anc 586 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
19 | 16, 18 | eqtrd 2856 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 ‘cfv 6355 Atomscatm 36414 HLchlt 36501 ⊥𝑃cpolN 37053 PSubClcpscN 37085 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-undef 7939 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-pmap 36655 df-polarityN 37054 df-psubclN 37086 |
This theorem is referenced by: osumcllem9N 37115 |
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