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Theorem oposlem 35338
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 2778 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2778 . . . . 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 35336 . . . 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 492 . . 3 (𝐾 ∈ OP → ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ))
12 fveq2 6446 . . . . . . 7 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1312eleq1d 2844 . . . . . 6 (𝑥 = 𝑋 → (( 𝑥) ∈ 𝐵 ↔ ( 𝑋) ∈ 𝐵))
14 2fveq3 6451 . . . . . . 7 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
15 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1614, 15eqeq12d 2793 . . . . . 6 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
17 breq1 4889 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
1812breq2d 4898 . . . . . . 7 (𝑥 = 𝑋 → (( 𝑦) ( 𝑥) ↔ ( 𝑦) ( 𝑋)))
1917, 18imbi12d 336 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦 → ( 𝑦) ( 𝑥)) ↔ (𝑋 𝑦 → ( 𝑦) ( 𝑋))))
2013, 16, 193anbi123d 1509 . . . . 5 (𝑥 = 𝑋 → ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ↔ (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋)))))
2115, 12oveq12d 6940 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2221eqeq1d 2780 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 1 ↔ (𝑋 ( 𝑋)) = 1 ))
2315, 12oveq12d 6940 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2423eqeq1d 2780 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 0 ↔ (𝑋 ( 𝑋)) = 0 ))
2520, 22, 243anbi123d 1509 . . . 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 4890 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
27 fveq2 6446 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2827breq1d 4896 . . . . . . 7 (𝑦 = 𝑌 → (( 𝑦) ( 𝑋) ↔ ( 𝑌) ( 𝑋)))
2926, 28imbi12d 336 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦 → ( 𝑦) ( 𝑋)) ↔ (𝑋 𝑌 → ( 𝑌) ( 𝑋))))
30293anbi3d 1515 . . . . 5 (𝑦 = 𝑌 → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑦 → ( 𝑦) ( 𝑋))) ↔ (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋)))))
31303anbi1d 1513 . . . 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 3524 . . 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 1104 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 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090   class class class wbr 4886  dom cdm 5355  cfv 6135  (class class class)co 6922  Basecbs 16255  lecple 16345  occoc 16346  Posetcpo 17326  lubclub 17328  glbcglb 17329  joincjn 17330  meetcmee 17331  0.cp0 17423  1.cp1 17424  OPcops 35328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-dm 5365  df-iota 6099  df-fv 6143  df-ov 6925  df-oposet 35332
This theorem is referenced by:  opoccl  35350  opococ  35351  oplecon3  35355  opexmid  35363  opnoncon  35364
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