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Mathbox for Norm Megill |
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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 35152 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | 1 | adantr 472 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
3 | simpr 479 | . . . 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 34975 | . . . . 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 35067 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) |
11 | 2, 3, 7, 10 | syl3anc 1477 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) |
12 | eqid 2760 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
13 | 12, 5, 8 | opoc1 34992 | . . . . 5 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
15 | 14 | breq1d 4814 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋) ↔ (0.‘𝐾)𝐶( ⊥ ‘𝑋))) |
16 | 4, 8 | opoccl 34984 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
17 | 1, 16 | sylan 489 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
18 | 17 | biantrurd 530 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((0.‘𝐾)𝐶( ⊥ ‘𝑋) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
19 | 11, 15, 18 | 3bitrd 294 | . 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 35076 | . . 3 ⊢ (𝐾 ∈ HL → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
22 | 21 | adantr 472 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
23 | 19, 22 | bitr4d 271 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 Basecbs 16059 occoc 16151 0.cp0 17238 1.cp1 17239 OPcops 34962 ⋖ ccvr 35052 Atomscatm 35053 HLchlt 35140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-p0 17240 df-p1 17241 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-hlat 35141 |
This theorem is referenced by: 1cvratex 35262 lhpoc 35803 |
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