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Theorem oml4 487
 Description: Orthomodular law. (Contributed by NM, 25-Oct-1997.)
Assertion
Ref Expression
oml4 ((ab) ∩ a) ≤ b

Proof of Theorem oml4
StepHypRef Expression
1 ancom 74 . . 3 ((ab) ∩ a) = (a ∩ (ab))
2 dfb 94 . . . . 5 (ab) = ((ab) ∪ (ab ))
32lan 77 . . . 4 (a ∩ (ab)) = (a ∩ ((ab) ∪ (ab )))
4 coman1 185 . . . . . . 7 (ab) C a
54comcom 453 . . . . . 6 a C (ab)
6 coman1 185 . . . . . . . . 9 (ab ) C a
76comcom 453 . . . . . . . 8 a C (ab )
87comcom2 183 . . . . . . 7 a C (ab )
98comcom5 458 . . . . . 6 a C (ab )
105, 9fh1 469 . . . . 5 (a ∩ ((ab) ∪ (ab ))) = ((a ∩ (ab)) ∪ (a ∩ (ab )))
11 or0 102 . . . . . 6 ((ab) ∪ 0) = (ab)
12 anidm 111 . . . . . . . . . 10 (aa) = a
1312ran 78 . . . . . . . . 9 ((aa) ∩ b) = (ab)
1413ax-r1 35 . . . . . . . 8 (ab) = ((aa) ∩ b)
15 anass 76 . . . . . . . 8 ((aa) ∩ b) = (a ∩ (ab))
1614, 15ax-r2 36 . . . . . . 7 (ab) = (a ∩ (ab))
17 ancom 74 . . . . . . . . 9 (b ∩ 0) = (0 ∩ b )
18 an0 108 . . . . . . . . 9 (b ∩ 0) = 0
19 dff 101 . . . . . . . . . 10 0 = (aa )
2019ran 78 . . . . . . . . 9 (0 ∩ b ) = ((aa ) ∩ b )
2117, 18, 203tr2 64 . . . . . . . 8 0 = ((aa ) ∩ b )
22 anass 76 . . . . . . . 8 ((aa ) ∩ b ) = (a ∩ (ab ))
2321, 22ax-r2 36 . . . . . . 7 0 = (a ∩ (ab ))
2416, 232or 72 . . . . . 6 ((ab) ∪ 0) = ((a ∩ (ab)) ∪ (a ∩ (ab )))
25 ancom 74 . . . . . 6 (ab) = (ba)
2611, 24, 253tr2 64 . . . . 5 ((a ∩ (ab)) ∪ (a ∩ (ab ))) = (ba)
2710, 26ax-r2 36 . . . 4 (a ∩ ((ab) ∪ (ab ))) = (ba)
283, 27ax-r2 36 . . 3 (a ∩ (ab)) = (ba)
291, 28ax-r2 36 . 2 ((ab) ∩ a) = (ba)
30 lea 160 . 2 (ba) ≤ b
3129, 30bltr 138 1 ((ab) ∩ a) ≤ b
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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