Theorem oplecon3b 39660

Description: Contraposition law for orthoposets. (chsscon3 31586 analog.) (Contributed by NM, 4-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
oplecon3b	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))

Proof of Theorem oplecon3b

Step	Hyp	Ref	Expression
1		opcon3.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		opcon3.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		opcon3.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
4	1, 2, 3	oplecon3 39659	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
5		simp1 1137	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
6	1, 3	opoccl 39654	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
7	6	3adant2 1132	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
8	1, 3	opoccl 39654	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
9	8	3adant3 1133	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
10	1, 2, 3	oplecon3 39659	. . . 4 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) ≤ ( ⊥ ‘( ⊥ ‘𝑌))))
11	5, 7, 9, 10	syl3anc 1374	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) ≤ ( ⊥ ‘( ⊥ ‘𝑌))))
12	1, 3	opococ 39655	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
13	12	3adant3 1133	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
14	1, 3	opococ 39655	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
15	14	3adant2 1132	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
16	13, 15	breq12d 5099	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑋)) ≤ ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑋 ≤ 𝑌))
17	11, 16	sylibd 239	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → 𝑋 ≤ 𝑌))
18	4, 17	impbid 212	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492  Basecbs 17170  lecple 17218  occoc 17219  OPcops 39632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-dm 5634  df-iota 6448  df-fv 6500  df-ov 7363  df-oposet 39636

This theorem is referenced by:  oplecon1b  39661  opltcon3b  39664  oldmm1  39677  omllaw4  39706  cvrcmp2  39744  glbconN  39837  lhpmod2i2  40498  lhpmod6i1  40499  lhprelat3N  40500  dochss  41825