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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 36492 | . . . . 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 36315 | . . . . 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 36407 | . . . 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 2821 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
13 | 12, 5, 8 | opoc1 36332 | . . . . 5 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘ 1 ) = (0.‘𝐾)) |
15 | 14 | breq1d 5068 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘ 1 )𝐶( ⊥ ‘𝑋) ↔ (0.‘𝐾)𝐶( ⊥ ‘𝑋))) |
16 | 4, 8 | opoccl 36324 | . . . . 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 36416 | . . 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 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 occoc 16567 0.cp0 17641 1.cp1 17642 OPcops 36302 ⋖ ccvr 36392 Atomscatm 36393 HLchlt 36480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-p0 17643 df-p1 17644 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-hlat 36481 |
This theorem is referenced by: 1cvratex 36603 lhpoc 37144 |
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