![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > opcon2b | Structured version Visualization version GIF version |
Description: Orthocomplement contraposition law. (negcon2 11560 analog.) (Contributed by NM, 16-Jan-2012.) |
Ref | Expression |
---|---|
opoccl.b | ⊢ 𝐵 = (Base‘𝐾) |
opoccl.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opcon2b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opoccl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | opoccl.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
3 | 1, 2 | opoccl 39176 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
4 | 3 | 3adant2 1130 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
5 | 1, 2 | opcon3b 39178 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) = ( ⊥ ‘𝑋))) |
6 | 4, 5 | syld3an3 1408 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) = ( ⊥ ‘𝑋))) |
7 | 1, 2 | opococ 39177 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
8 | 7 | 3adant2 1130 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
9 | 8 | eqeq1d 2737 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑌)) = ( ⊥ ‘𝑋) ↔ 𝑌 = ( ⊥ ‘𝑋))) |
10 | 6, 9 | bitrd 279 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ( ⊥ ‘𝑌) ↔ 𝑌 = ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Basecbs 17245 occoc 17306 OPcops 39154 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-oposet 39158 |
This theorem is referenced by: opcon1b 39180 riotaocN 39191 glbconxN 39361 |
Copyright terms: Public domain | W3C validator |