Theorem oplecon3 37734

Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
opcon3.b	⊢ 𝐵 = (Base‘𝐾)
opcon3.l	⊢ ≤ = (le‘𝐾)
opcon3.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
oplecon3	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))

Proof of Theorem oplecon3

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 2731	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2731	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
6		eqid 2731	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		eqid 2731	. . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 37717	. . 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)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5110  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  lecple 17154  occoc 17155  joincjn 18214  meetcmee 18215  0.cp0 18326  1.cp1 18327  OPcops 37707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-nul 5268

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-dm 5648  df-iota 6453  df-fv 6509  df-ov 7365  df-oposet 37711

This theorem is referenced by:  oplecon3b  37735