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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cvrco | Structured version Visualization version GIF version | ||
| Description: The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.) | 
| Ref | Expression | 
|---|---|
| 1cvrco.b | ⊢ 𝐵 = (Base‘𝐾) | 
| 1cvrco.u | ⊢ 1 = (1.‘𝐾) | 
| 1cvrco.o | ⊢ ⊥ = (oc‘𝐾) | 
| 1cvrco.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) | 
| 1cvrco.a | ⊢ 𝐴 = (Atoms‘𝐾) | 
| Ref | Expression | 
|---|---|
| 1cvrco | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hlop 39364 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
| 2 | 1 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | 
| 3 | simpr 484 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 4 | 1cvrco.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 5 | 1cvrco.u | . . . . . 6 ⊢ 1 = (1.‘𝐾) | |
| 6 | 4, 5 | op1cl 39187 | . . . . 5 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) | 
| 7 | 2, 6 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 1 ∈ 𝐵) | 
| 8 | 1cvrco.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
| 9 | 1cvrco.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
| 10 | 4, 8, 9 | cvrcon3b 39279 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) | 
| 11 | 2, 3, 7, 10 | syl3anc 1372 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) | 
| 12 | eqid 2736 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
| 13 | 12, 5, 8 | opoc1 39204 | . . . . 5 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = (0.‘𝐾)) | 
| 14 | 2, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘ 1 ) = (0.‘𝐾)) | 
| 15 | 14 | breq1d 5152 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋) ↔ (0.‘𝐾)𝐶( ⊥ ‘𝑋))) | 
| 16 | 4, 8 | opoccl 39196 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) | 
| 17 | 1, 16 | sylan 580 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) | 
| 18 | 17 | biantrurd 532 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((0.‘𝐾)𝐶( ⊥ ‘𝑋) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) | 
| 19 | 11, 15, 18 | 3bitrd 305 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) | 
| 20 | 1cvrco.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 21 | 4, 12, 9, 20 | isat 39288 | . . 3 ⊢ (𝐾 ∈ HL → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) | 
| 22 | 21 | adantr 480 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) | 
| 23 | 19, 22 | bitr4d 282 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 Basecbs 17248 occoc 17306 0.cp0 18469 1.cp1 18470 OPcops 39174 ⋖ ccvr 39264 Atomscatm 39265 HLchlt 39352 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-p0 18471 df-p1 18472 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-hlat 39353 | 
| This theorem is referenced by: 1cvratex 39476 lhpoc 40017 | 
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