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Theorem omllaw 35318
Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
omllaw.b 𝐵 = (Base‘𝐾)
omllaw.l = (le‘𝐾)
omllaw.j = (join‘𝐾)
omllaw.m = (meet‘𝐾)
omllaw.o = (oc‘𝐾)
Assertion
Ref Expression
omllaw ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))

Proof of Theorem omllaw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omllaw.b . . . . 5 𝐵 = (Base‘𝐾)
2 omllaw.l . . . . 5 = (le‘𝐾)
3 omllaw.j . . . . 5 = (join‘𝐾)
4 omllaw.m . . . . 5 = (meet‘𝐾)
5 omllaw.o . . . . 5 = (oc‘𝐾)
61, 2, 3, 4, 5isoml 35313 . . . 4 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
76simprbi 492 . . 3 (𝐾 ∈ OML → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))
8 breq1 4876 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
10 fveq2 6433 . . . . . . . 8 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1110oveq2d 6921 . . . . . . 7 (𝑥 = 𝑋 → (𝑦 ( 𝑥)) = (𝑦 ( 𝑋)))
129, 11oveq12d 6923 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 ( 𝑥))) = (𝑋 (𝑦 ( 𝑋))))
1312eqeq2d 2835 . . . . 5 (𝑥 = 𝑋 → (𝑦 = (𝑥 (𝑦 ( 𝑥))) ↔ 𝑦 = (𝑋 (𝑦 ( 𝑋)))))
148, 13imbi12d 336 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))) ↔ (𝑋 𝑦𝑦 = (𝑋 (𝑦 ( 𝑋))))))
15 breq2 4877 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
16 id 22 . . . . . 6 (𝑦 = 𝑌𝑦 = 𝑌)
17 oveq1 6912 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 ( 𝑋)) = (𝑌 ( 𝑋)))
1817oveq2d 6921 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 ( 𝑋))) = (𝑋 (𝑌 ( 𝑋))))
1916, 18eqeq12d 2840 . . . . 5 (𝑦 = 𝑌 → (𝑦 = (𝑋 (𝑦 ( 𝑋))) ↔ 𝑌 = (𝑋 (𝑌 ( 𝑋)))))
2015, 19imbi12d 336 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 = (𝑋 (𝑦 ( 𝑋)))) ↔ (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋))))))
2114, 20rspc2v 3539 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋))))))
227, 21syl5com 31 . 2 (𝐾 ∈ OML → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋))))))
23223impib 1150 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  occoc 16313  joincjn 17297  meetcmee 17298  OLcol 35249  OMLcoml 35250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-oml 35254
This theorem is referenced by:  omllaw2N  35319  omllaw3  35320  omllaw4  35321
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