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Theorem woml6 436
 Description: Variant of weakly orthomodular law.
Assertion
Ref Expression
woml6 ((a1 b) ∪ (a2 b)) = 1

Proof of Theorem woml6
StepHypRef Expression
1 df-i1 44 . . . . . 6 (a1 b) = (a ∪ (ab))
2 df-a 40 . . . . . . 7 (ab) = (ab )
32lor 70 . . . . . 6 (a ∪ (ab)) = (a ∪ (ab ) )
41, 3ax-r2 36 . . . . 5 (a1 b) = (a ∪ (ab ) )
54ax-r4 37 . . . 4 (a1 b) = (a ∪ (ab ) )
6 df-a 40 . . . . 5 (a ∩ (ab )) = (a ∪ (ab ) )
76ax-r1 35 . . . 4 (a ∪ (ab ) ) = (a ∩ (ab ))
85, 7ax-r2 36 . . 3 (a1 b) = (a ∩ (ab ))
9 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
108, 92or 72 . 2 ((a1 b) ∪ (a2 b)) = ((a ∩ (ab )) ∪ (b ∪ (ab )))
11 ax-a2 31 . . . . 5 ((a ∩ (ab )) ∪ b) = (b ∪ (a ∩ (ab )))
12 ancom 74 . . . . . 6 (a ∩ (ab )) = ((ab ) ∩ a)
1312lor 70 . . . . 5 (b ∪ (a ∩ (ab ))) = (b ∪ ((ab ) ∩ a))
1411, 13ax-r2 36 . . . 4 ((a ∩ (ab )) ∪ b) = (b ∪ ((ab ) ∩ a))
1514ax-r5 38 . . 3 (((a ∩ (ab )) ∪ b) ∪ (ab )) = ((b ∪ ((ab ) ∩ a)) ∪ (ab ))
16 ax-a3 32 . . 3 (((a ∩ (ab )) ∪ b) ∪ (ab )) = ((a ∩ (ab )) ∪ (b ∪ (ab )))
17 1b 117 . . . . 5 (1 ≡ ((b ∪ ((ab ) ∩ a)) ∪ (ab ))) = ((b ∪ ((ab ) ∩ a)) ∪ (ab ))
1817ax-r1 35 . . . 4 ((b ∪ ((ab ) ∩ a)) ∪ (ab )) = (1 ≡ ((b ∪ ((ab ) ∩ a)) ∪ (ab )))
19 wcomorr 412 . . . . . . . . . . . 12 C (b , (ba )) = 1
20 ax-a2 31 . . . . . . . . . . . . 13 (ba ) = (ab )
2120bi1 118 . . . . . . . . . . . 12 ((ba ) ≡ (ab )) = 1
2219, 21wcbtr 411 . . . . . . . . . . 11 C (b , (ab )) = 1
2322wcomcom 414 . . . . . . . . . 10 C ((ab ), b ) = 1
2423wcomcom3 416 . . . . . . . . 9 C ((ab ) , b ) = 1
2524wcomcom5 420 . . . . . . . 8 C ((ab ), b) = 1
26 wcomorr 412 . . . . . . . . . . 11 C (a , (ab )) = 1
2726wcomcom 414 . . . . . . . . . 10 C ((ab ), a ) = 1
2827wcomcom3 416 . . . . . . . . 9 C ((ab ) , a ) = 1
2928wcomcom5 420 . . . . . . . 8 C ((ab ), a) = 1
3025, 29wfh4 426 . . . . . . 7 ((b ∪ ((ab ) ∩ a)) ≡ ((b ∪ (ab )) ∩ (ba))) = 1
3130wr5-2v 366 . . . . . 6 (((b ∪ ((ab ) ∩ a)) ∪ (ab )) ≡ (((b ∪ (ab )) ∩ (ba)) ∪ (ab ))) = 1
32 or12 80 . . . . . . . . . . . . 13 (b ∪ (ab )) = (a ∪ (bb ))
33 df-t 41 . . . . . . . . . . . . . . 15 1 = (bb )
3433lor 70 . . . . . . . . . . . . . 14 (a ∪ 1) = (a ∪ (bb ))
3534ax-r1 35 . . . . . . . . . . . . 13 (a ∪ (bb )) = (a ∪ 1)
36 or1 104 . . . . . . . . . . . . 13 (a ∪ 1) = 1
3732, 35, 363tr 65 . . . . . . . . . . . 12 (b ∪ (ab )) = 1
3837ran 78 . . . . . . . . . . 11 ((b ∪ (ab )) ∩ (ba)) = (1 ∩ (ba))
39 ancom 74 . . . . . . . . . . 11 (1 ∩ (ba)) = ((ba) ∩ 1)
4038, 39ax-r2 36 . . . . . . . . . 10 ((b ∪ (ab )) ∩ (ba)) = ((ba) ∩ 1)
41 an1 106 . . . . . . . . . 10 ((ba) ∩ 1) = (ba)
42 ax-a2 31 . . . . . . . . . 10 (ba) = (ab)
4340, 41, 423tr 65 . . . . . . . . 9 ((b ∪ (ab )) ∩ (ba)) = (ab)
44 anor3 90 . . . . . . . . 9 (ab ) = (ab)
4543, 442or 72 . . . . . . . 8 (((b ∪ (ab )) ∩ (ba)) ∪ (ab )) = ((ab) ∪ (ab) )
46 df-t 41 . . . . . . . . 9 1 = ((ab) ∪ (ab) )
4746ax-r1 35 . . . . . . . 8 ((ab) ∪ (ab) ) = 1
4845, 47ax-r2 36 . . . . . . 7 (((b ∪ (ab )) ∩ (ba)) ∪ (ab )) = 1
4948bi1 118 . . . . . 6 ((((b ∪ (ab )) ∩ (ba)) ∪ (ab )) ≡ 1) = 1
5031, 49wr2 371 . . . . 5 (((b ∪ ((ab ) ∩ a)) ∪ (ab )) ≡ 1) = 1
5150wr1 197 . . . 4 (1 ≡ ((b ∪ ((ab ) ∩ a)) ∪ (ab ))) = 1
5218, 51ax-r2 36 . . 3 ((b ∪ ((ab ) ∩ a)) ∪ (ab )) = 1
5315, 16, 523tr2 64 . 2 ((a ∩ (ab )) ∪ (b ∪ (ab ))) = 1
5410, 53ax-r2 36 1 ((a1 b) ∪ (a2 b)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   →2 wi2 13 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-wom 361 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by:  lem3.4.1  1075
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