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Theorem oposlem 36378
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 2824 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
3 eqid 2824 . . . . 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 36376 . . . 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 500 . . 3 (𝐾 ∈ OP → ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ))
12 fveq2 6651 . . . . . . 7 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1312eleq1d 2900 . . . . . 6 (𝑥 = 𝑋 → (( 𝑥) ∈ 𝐵 ↔ ( 𝑋) ∈ 𝐵))
14 2fveq3 6656 . . . . . . 7 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
15 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1614, 15eqeq12d 2840 . . . . . 6 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
17 breq1 5050 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
1812breq2d 5059 . . . . . . 7 (𝑥 = 𝑋 → (( 𝑦) ( 𝑥) ↔ ( 𝑦) ( 𝑋)))
1917, 18imbi12d 348 . . . . . 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 7156 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2221eqeq1d 2826 . . . . 5 (𝑥 = 𝑋 → ((𝑥 ( 𝑥)) = 1 ↔ (𝑋 ( 𝑋)) = 1 ))
2315, 12oveq12d 7156 . . . . . 6 (𝑥 = 𝑋 → (𝑥 ( 𝑥)) = (𝑋 ( 𝑋)))
2423eqeq1d 2826 . . . . 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 5051 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
27 fveq2 6651 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2827breq1d 5057 . . . . . . 7 (𝑦 = 𝑌 → (( 𝑦) ( 𝑋) ↔ ( 𝑌) ( 𝑋)))
2926, 28imbi12d 348 . . . . . 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 3618 . . 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 510 . 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132   class class class wbr 5047  dom cdm 5536  cfv 6336  (class class class)co 7138  Basecbs 16472  lecple 16561  occoc 16562  Posetcpo 17539  lubclub 17541  glbcglb 17542  joincjn 17543  meetcmee 17544  0.cp0 17636  1.cp1 17637  OPcops 36368
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5191
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-dm 5546  df-iota 6295  df-fv 6344  df-ov 7141  df-oposet 36372
This theorem is referenced by:  opoccl  36390  opococ  36391  oplecon3  36395  opexmid  36403  opnoncon  36404
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