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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 37111 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
3 | simpr 488 | . . . 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 36934 | . . . . 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 37026 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) |
11 | 2, 3, 7, 10 | syl3anc 1373 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) |
12 | eqid 2737 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
13 | 12, 5, 8 | opoc1 36951 | . . . . 5 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
15 | 14 | breq1d 5060 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋) ↔ (0.‘𝐾)𝐶( ⊥ ‘𝑋))) |
16 | 4, 8 | opoccl 36943 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
17 | 1, 16 | sylan 583 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
18 | 17 | biantrurd 536 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((0.‘𝐾)𝐶( ⊥ ‘𝑋) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
19 | 11, 15, 18 | 3bitrd 308 | . 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 37035 | . . 3 ⊢ (𝐾 ∈ HL → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
22 | 21 | adantr 484 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
23 | 19, 22 | bitr4d 285 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5050 ‘cfv 6377 Basecbs 16757 occoc 16807 0.cp0 17926 1.cp1 17927 OPcops 36921 ⋖ ccvr 37011 Atomscatm 37012 HLchlt 37099 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-proset 17799 df-poset 17817 df-plt 17833 df-lub 17849 df-glb 17850 df-p0 17928 df-p1 17929 df-oposet 36925 df-ol 36927 df-oml 36928 df-covers 37015 df-ats 37016 df-hlat 37100 |
This theorem is referenced by: 1cvratex 37222 lhpoc 37763 |
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