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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 36500 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | 1 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
3 | simpr 487 | . . . 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 36323 | . . . . 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 36415 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) |
11 | 2, 3, 7, 10 | syl3anc 1367 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋))) |
12 | eqid 2823 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
13 | 12, 5, 8 | opoc1 36340 | . . . . 5 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
15 | 14 | breq1d 5078 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋) ↔ (0.‘𝐾)𝐶( ⊥ ‘𝑋))) |
16 | 4, 8 | opoccl 36332 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
17 | 1, 16 | sylan 582 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
18 | 17 | biantrurd 535 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((0.‘𝐾)𝐶( ⊥ ‘𝑋) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
19 | 11, 15, 18 | 3bitrd 307 | . 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 36424 | . . 3 ⊢ (𝐾 ∈ HL → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
22 | 21 | adantr 483 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ (0.‘𝐾)𝐶( ⊥ ‘𝑋)))) |
23 | 19, 22 | bitr4d 284 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑋𝐶 1 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 occoc 16575 0.cp0 17649 1.cp1 17650 OPcops 36310 ⋖ ccvr 36400 Atomscatm 36401 HLchlt 36488 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-p0 17651 df-p1 17652 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-hlat 36489 |
This theorem is referenced by: 1cvratex 36611 lhpoc 37152 |
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