Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opcon3b Structured version   Visualization version   GIF version

Theorem opcon3b 36331
Description: Contraposition law for orthoposets. (chcon3i 29242 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
opoccl.b 𝐵 = (Base‘𝐾)
opoccl.o = (oc‘𝐾)
Assertion
Ref Expression
opcon3b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))

Proof of Theorem opcon3b
StepHypRef Expression
1 fveq2 6669 . . 3 (𝑌 = 𝑋 → ( 𝑌) = ( 𝑋))
21eqcoms 2829 . 2 (𝑋 = 𝑌 → ( 𝑌) = ( 𝑋))
3 fveq2 6669 . . . 4 (( 𝑋) = ( 𝑌) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
43eqcoms 2829 . . 3 (( 𝑌) = ( 𝑋) → ( ‘( 𝑋)) = ( ‘( 𝑌)))
5 opoccl.b . . . . . 6 𝐵 = (Base‘𝐾)
6 opoccl.o . . . . . 6 = (oc‘𝐾)
75, 6opococ 36330 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
873adant3 1128 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
95, 6opococ 36330 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1093adant2 1127 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
118, 10eqeq12d 2837 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) = ( ‘( 𝑌)) ↔ 𝑋 = 𝑌))
124, 11syl5ib 246 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) = ( 𝑋) → 𝑋 = 𝑌))
132, 12impbid2 228 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1083   = wceq 1533  wcel 2110  cfv 6354  Basecbs 16482  occoc 16572  OPcops 36307
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-dm 5564  df-iota 6313  df-fv 6362  df-ov 7158  df-oposet 36311
This theorem is referenced by:  opcon2b  36332  omllaw4  36381  cmtbr2N  36388  cvrcmp2  36419  lhpmod2i2  37173  lhpmod6i1  37174
  Copyright terms: Public domain W3C validator