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

Theorem oposlem 35138
Description: Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
Hypotheses
Ref Expression
oposlem.b 𝐵 = (Base‘𝐾)
oposlem.l = (le‘𝐾)
oposlem.o = (oc‘𝐾)
oposlem.j = (join‘𝐾)
oposlem.m = (meet‘𝐾)
oposlem.f 0 = (0.‘𝐾)
oposlem.u 1 = (1.‘𝐾)
Assertion
Ref Expression
oposlem ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))

Proof of Theorem oposlem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oposlem.b . . . . 5 𝐵 = (Base‘𝐾)
2 eqid 2765 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2765 . . . . 5 (glb‘𝐾) = (glb‘𝐾)
4 oposlem.l . . . . 5 = (le‘𝐾)
5 oposlem.o . . . . 5 = (oc‘𝐾)
6 oposlem.j . . . . 5 = (join‘𝐾)
7 oposlem.m . . . . 5 = (meet‘𝐾)
8 oposlem.f . . . . 5 0 = (0.‘𝐾)
9 oposlem.u . . . . 5 1 = (1.‘𝐾)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 35136 . . . 4 (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾)) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
1110simprbi 490 . . 3 (𝐾 ∈ OP → ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ))
12 fveq2 6375 . . . . . . 7 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1312eleq1d 2829 . . . . . 6 (𝑥 = 𝑋 → (( 𝑥) ∈ 𝐵 ↔ ( 𝑋) ∈ 𝐵))
14 2fveq3 6380 . . . . . . 7 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
15 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1614, 15eqeq12d 2780 . . . . . 6 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
17 breq1 4812 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
1812breq2d 4821 . . . . . . 7 (𝑥 = 𝑋 → (( 𝑦) ( 𝑥) ↔ ( 𝑦) ( 𝑋)))
1917, 18imbi12d 335 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦 → ( 𝑦) ( 𝑥)) ↔ (𝑋 𝑦 → ( 𝑦) ( 𝑋))))
2013, 16, 193anbi123d 1560 . . . . 5 (𝑥 = 𝑋 → ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ↔ (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋)))))
2115, 12oveq12d 6860 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2221eqeq1d 2767 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 1 ↔ (𝑋 ( 𝑋)) = 1 ))
2315, 12oveq12d 6860 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2423eqeq1d 2767 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 0 ↔ (𝑋 ( 𝑋)) = 0 ))
2520, 22, 243anbi123d 1560 . . . 4 (𝑥 = 𝑋 → (((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ) ↔ ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 )))
26 breq2 4813 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
27 fveq2 6375 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2827breq1d 4819 . . . . . . 7 (𝑦 = 𝑌 → (( 𝑦) ( 𝑋) ↔ ( 𝑌) ( 𝑋)))
2926, 28imbi12d 335 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦 → ( 𝑦) ( 𝑋)) ↔ (𝑋 𝑌 → ( 𝑌) ( 𝑋))))
30293anbi3d 1566 . . . . 5 (𝑦 = 𝑌 → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋))) ↔ (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋)))))
31303anbi1d 1564 . . . 4 (𝑦 = 𝑌 → (((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ) ↔ ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 )))
3225, 31rspc2v 3474 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 )))
3311, 32mpan9 502 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵)) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))
34333impb 1143 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055   class class class wbr 4809  dom cdm 5277  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  occoc 16222  Posetcpo 17206  lubclub 17208  glbcglb 17209  joincjn 17210  meetcmee 17211  0.cp0 17303  1.cp1 17304  OPcops 35128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4949
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-dm 5287  df-iota 6031  df-fv 6076  df-ov 6845  df-oposet 35132
This theorem is referenced by:  opoccl  35150  opococ  35151  oplecon3  35155  opexmid  35163  opnoncon  35164
  Copyright terms: Public domain W3C validator