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Theorem omllaw 37708
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 37703 . . . 4 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
76simprbi 498 . . 3 (𝐾 ∈ OML β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))))
8 breq1 5109 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
9 id 22 . . . . . . 7 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
10 fveq2 6843 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
1110oveq2d 7374 . . . . . . 7 (π‘₯ = 𝑋 β†’ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)) = (𝑦 ∧ ( βŠ₯ β€˜π‘‹)))
129, 11oveq12d 7376 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))) = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))))
1312eqeq2d 2748 . . . . 5 (π‘₯ = 𝑋 β†’ (𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))) ↔ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹)))))
148, 13imbi12d 345 . . . 4 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))) ↔ (𝑋 ≀ 𝑦 β†’ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))))))
15 breq2 5110 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
16 id 22 . . . . . 6 (𝑦 = π‘Œ β†’ 𝑦 = π‘Œ)
17 oveq1 7365 . . . . . . 7 (𝑦 = π‘Œ β†’ (𝑦 ∧ ( βŠ₯ β€˜π‘‹)) = (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))
1817oveq2d 7374 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))) = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1916, 18eqeq12d 2753 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 = (𝑋 ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘‹))) ↔ π‘Œ = (𝑋 ∨ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
2015, 19imbi12d 345 . . . 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 3591 . . 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 1117 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  occoc 17142  joincjn 18201  meetcmee 18202  OLcol 37639  OMLcoml 37640
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
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-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-iota 6449  df-fv 6505  df-ov 7361  df-oml 37644
This theorem is referenced by:  omllaw2N  37709  omllaw3  37710  omllaw4  37711
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