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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 2735 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2735 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2735 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | eqid 2735 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 39164 | . . 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 1141 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))) |
10 | 9 | simp3d 1143 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 occoc 17306 joincjn 18369 meetcmee 18370 0.cp0 18481 1.cp1 18482 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: oplecon3b 39182 |
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