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Theorem omllaw 38624
Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
omllaw.b 𝐡 = (Baseβ€˜πΎ)
omllaw.l ≀ = (leβ€˜πΎ)
omllaw.j ∨ = (joinβ€˜πΎ)
omllaw.m ∧ = (meetβ€˜πΎ)
omllaw.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
omllaw ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))

Proof of Theorem omllaw
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omllaw.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 omllaw.l . . . . 5 ≀ = (leβ€˜πΎ)
3 omllaw.j . . . . 5 ∨ = (joinβ€˜πΎ)
4 omllaw.m . . . . 5 ∧ = (meetβ€˜πΎ)
5 omllaw.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
61, 2, 3, 4, 5isoml 38619 . . . 4 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
76simprbi 496 . . 3 (𝐾 ∈ OML β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))))
8 breq1 5144 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
9 id 22 . . . . . . 7 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
10 fveq2 6884 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
1110oveq2d 7420 . . . . . . 7 (π‘₯ = 𝑋 β†’ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)) = (𝑦 ∧ ( βŠ₯ β€˜π‘‹)))
129, 11oveq12d 7422 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))) = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))))
1312eqeq2d 2737 . . . . 5 (π‘₯ = 𝑋 β†’ (𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))) ↔ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹)))))
148, 13imbi12d 344 . . . 4 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))) ↔ (𝑋 ≀ 𝑦 β†’ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))))))
15 breq2 5145 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
16 id 22 . . . . . 6 (𝑦 = π‘Œ β†’ 𝑦 = π‘Œ)
17 oveq1 7411 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑦 ∧ ( βŠ₯ β€˜π‘‹)) = (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))
1817oveq2d 7420 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))) = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1916, 18eqeq12d 2742 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))) ↔ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
2015, 19imbi12d 344 . . . 4 (𝑦 = π‘Œ β†’ ((𝑋 ≀ 𝑦 β†’ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹)))) ↔ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))))
2114, 20rspc2v 3617 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))) β†’ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))))
227, 21syl5com 31 . 2 (𝐾 ∈ OML β†’ ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))))
23223impib 1113 1 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  occoc 17212  joincjn 18274  meetcmee 18275  OLcol 38555  OMLcoml 38556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-oml 38560
This theorem is referenced by:  omllaw2N  38625  omllaw3  38626  omllaw4  38627
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