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Theorem oposlem 37647
Description: Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
Hypotheses
Ref Expression
oposlem.b 𝐡 = (Baseβ€˜πΎ)
oposlem.l ≀ = (leβ€˜πΎ)
oposlem.o βŠ₯ = (ocβ€˜πΎ)
oposlem.j ∨ = (joinβ€˜πΎ)
oposlem.m ∧ = (meetβ€˜πΎ)
oposlem.f 0 = (0.β€˜πΎ)
oposlem.u 1 = (1.β€˜πΎ)
Assertion
Ref Expression
oposlem ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ))

Proof of Theorem oposlem
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oposlem.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 eqid 2737 . . . . 5 (lubβ€˜πΎ) = (lubβ€˜πΎ)
3 eqid 2737 . . . . 5 (glbβ€˜πΎ) = (glbβ€˜πΎ)
4 oposlem.l . . . . 5 ≀ = (leβ€˜πΎ)
5 oposlem.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
6 oposlem.j . . . . 5 ∨ = (joinβ€˜πΎ)
7 oposlem.m . . . . 5 ∧ = (meetβ€˜πΎ)
8 oposlem.f . . . . 5 0 = (0.β€˜πΎ)
9 oposlem.u . . . . 5 1 = (1.β€˜πΎ)
101, 2, 3, 4, 5, 6, 7, 8, 9isopos 37645 . . . 4 (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐡 ∈ dom (lubβ€˜πΎ) ∧ 𝐡 ∈ dom (glbβ€˜πΎ)) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((( βŠ₯ β€˜π‘₯) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ∧ (π‘₯ ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = 1 ∧ (π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = 0 )))
1110simprbi 498 . . 3 (𝐾 ∈ OP β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((( βŠ₯ β€˜π‘₯) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ∧ (π‘₯ ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = 1 ∧ (π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = 0 ))
12 fveq2 6843 . . . . . . 7 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
1312eleq1d 2823 . . . . . 6 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜π‘₯) ∈ 𝐡 ↔ ( βŠ₯ β€˜π‘‹) ∈ 𝐡))
14 2fveq3 6848 . . . . . . 7 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
15 id 22 . . . . . . 7 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
1614, 15eqeq12d 2753 . . . . . 6 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
17 breq1 5109 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
1812breq2d 5118 . . . . . . 7 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯) ↔ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹)))
1917, 18imbi12d 345 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯)) ↔ (𝑋 ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹))))
2013, 16, 193anbi123d 1437 . . . . 5 (π‘₯ = 𝑋 β†’ ((( βŠ₯ β€˜π‘₯) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ∧ (π‘₯ ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯))) ↔ (( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹)))))
2115, 12oveq12d 7376 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = (𝑋 ∨ ( βŠ₯ β€˜π‘‹)))
2221eqeq1d 2739 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = 1 ↔ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ))
2315, 12oveq12d 7376 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = (𝑋 ∧ ( βŠ₯ β€˜π‘‹)))
2423eqeq1d 2739 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = 0 ↔ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ))
2520, 22, 243anbi123d 1437 . . . 4 (π‘₯ = 𝑋 β†’ (((( βŠ₯ β€˜π‘₯) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ∧ (π‘₯ ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = 1 ∧ (π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = 0 ) ↔ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 )))
26 breq2 5110 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
27 fveq2 6843 . . . . . . . 8 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜π‘¦) = ( βŠ₯ β€˜π‘Œ))
2827breq1d 5116 . . . . . . 7 (𝑦 = π‘Œ β†’ (( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹) ↔ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))
2926, 28imbi12d 345 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑋 ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹)) ↔ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))))
30293anbi3d 1443 . . . . 5 (𝑦 = π‘Œ β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹))) ↔ (( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))))
31303anbi1d 1441 . . . 4 (𝑦 = π‘Œ β†’ (((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) ↔ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 )))
3225, 31rspc2v 3591 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((( βŠ₯ β€˜π‘₯) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ∧ (π‘₯ ≀ 𝑦 β†’ ( βŠ₯ β€˜π‘¦) ≀ ( βŠ₯ β€˜π‘₯))) ∧ (π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = 1 ∧ (π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = 0 ) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 )))
3311, 32mpan9 508 . 2 ((𝐾 ∈ OP ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ))
34333impb 1116 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹))) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘‹)) = 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5106  dom cdm 5634  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  occoc 17142  Posetcpo 18197  lubclub 18199  glbcglb 18200  joincjn 18201  meetcmee 18202  0.cp0 18313  1.cp1 18314  OPcops 37637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361  df-oposet 37641
This theorem is referenced by:  opoccl  37659  opococ  37660  oplecon3  37664  opexmid  37672  opnoncon  37673
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