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Theorem oposlem 38880
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 2726 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2726 . . . . 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 38878 . . . 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 495 . . 3 (𝐾 ∈ OP → ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ))
12 fveq2 6901 . . . . . . 7 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1312eleq1d 2811 . . . . . 6 (𝑥 = 𝑋 → (( 𝑥) ∈ 𝐵 ↔ ( 𝑋) ∈ 𝐵))
14 2fveq3 6906 . . . . . . 7 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
15 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1614, 15eqeq12d 2742 . . . . . 6 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
17 breq1 5156 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
1812breq2d 5165 . . . . . . 7 (𝑥 = 𝑋 → (( 𝑦) ( 𝑥) ↔ ( 𝑦) ( 𝑋)))
1917, 18imbi12d 343 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦 → ( 𝑦) ( 𝑥)) ↔ (𝑋 𝑦 → ( 𝑦) ( 𝑋))))
2013, 16, 193anbi123d 1433 . . . . 5 (𝑥 = 𝑋 → ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ↔ (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋)))))
2115, 12oveq12d 7442 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2221eqeq1d 2728 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 1 ↔ (𝑋 ( 𝑋)) = 1 ))
2315, 12oveq12d 7442 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2423eqeq1d 2728 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 0 ↔ (𝑋 ( 𝑋)) = 0 ))
2520, 22, 243anbi123d 1433 . . . 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 5157 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
27 fveq2 6901 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2827breq1d 5163 . . . . . . 7 (𝑦 = 𝑌 → (( 𝑦) ( 𝑋) ↔ ( 𝑌) ( 𝑋)))
2926, 28imbi12d 343 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦 → ( 𝑦) ( 𝑋)) ↔ (𝑋 𝑌 → ( 𝑌) ( 𝑋))))
30293anbi3d 1439 . . . . 5 (𝑦 = 𝑌 → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋))) ↔ (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋)))))
31303anbi1d 1437 . . . 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 3619 . . 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 505 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵)) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋 ( 𝑋)) = 1 ∧ (𝑋 ( 𝑋)) = 0 ))
34333impb 1112 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 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051   class class class wbr 5153  dom cdm 5682  cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  occoc 17274  Posetcpo 18332  lubclub 18334  glbcglb 18335  joincjn 18336  meetcmee 18337  0.cp0 18448  1.cp1 18449  OPcops 38870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-dm 5692  df-iota 6506  df-fv 6562  df-ov 7427  df-oposet 38874
This theorem is referenced by:  opoccl  38892  opococ  38893  oplecon3  38897  opexmid  38905  opnoncon  38906
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