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Theorem oposlem 38052
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 2733 . . . . 5 (lubβ€˜πΎ) = (lubβ€˜πΎ)
3 eqid 2733 . . . . 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 38050 . . . 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 6892 . . . . . . 7 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
1312eleq1d 2819 . . . . . 6 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜π‘₯) ∈ 𝐡 ↔ ( βŠ₯ β€˜π‘‹) ∈ 𝐡))
14 2fveq3 6897 . . . . . . 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 2749 . . . . . 6 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯ ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋))
17 breq1 5152 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
1812breq2d 5161 . . . . . . 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 7427 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = (𝑋 ∨ ( βŠ₯ β€˜π‘‹)))
2221eqeq1d 2735 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘₯ ∨ ( βŠ₯ β€˜π‘₯)) = 1 ↔ (𝑋 ∨ ( βŠ₯ β€˜π‘‹)) = 1 ))
2315, 12oveq12d 7427 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ∧ ( βŠ₯ β€˜π‘₯)) = (𝑋 ∧ ( βŠ₯ β€˜π‘‹)))
2423eqeq1d 2735 . . . . 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 5153 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
27 fveq2 6892 . . . . . . . 8 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜π‘¦) = ( βŠ₯ β€˜π‘Œ))
2827breq1d 5159 . . . . . . 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 3623 . . 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 3062   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  occoc 17205  Posetcpo 18260  lubclub 18262  glbcglb 18263  joincjn 18264  meetcmee 18265  0.cp0 18376  1.cp1 18377  OPcops 38042
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 2704  ax-nul 5307
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 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412  df-oposet 38046
This theorem is referenced by:  opoccl  38064  opococ  38065  oplecon3  38069  opexmid  38077  opnoncon  38078
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