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| Mirrors > Home > MPE Home > Th. List > Mathboxes > oplecon3 | Structured version Visualization version GIF version | ||
| Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.) | 
| Ref | Expression | 
|---|---|
| opcon3.b | ⊢ 𝐵 = (Base‘𝐾) | 
| opcon3.l | ⊢ ≤ = (le‘𝐾) | 
| opcon3.o | ⊢ ⊥ = (oc‘𝐾) | 
| Ref | Expression | 
|---|---|
| oplecon3 | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opcon3.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | opcon3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 3 | opcon3.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
| 4 | eqid 2736 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 5 | eqid 2736 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 6 | eqid 2736 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
| 7 | eqid 2736 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 39184 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾))) | 
| 9 | 8 | simp1d 1142 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))) | 
| 10 | 9 | simp3d 1144 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 lecple 17305 occoc 17306 joincjn 18358 meetcmee 18359 0.cp0 18469 1.cp1 18470 OPcops 39174 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435 df-oposet 39178 | 
| This theorem is referenced by: oplecon3b 39202 | 
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