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Theorem wcomlem 382
 Description: Analogue of commutation is symmetric. Similar to Kalmbach 83 p. 22. (Contributed by NM, 27-Jan-2002.)
Hypothesis
Ref Expression
wcomlem.1 (a ≡ ((ab) ∪ (ab ))) = 1
Assertion
Ref Expression
wcomlem (b ≡ ((ba) ∪ (ba ))) = 1

Proof of Theorem wcomlem
StepHypRef Expression
1 ax-a2 31 . . . . . . . . . 10 (ab) = (ba )
21bi1 118 . . . . . . . . 9 ((ab) ≡ (ba )) = 1
32wran 369 . . . . . . . 8 (((ab) ∩ b) ≡ ((ba ) ∩ b)) = 1
4 ancom 74 . . . . . . . . 9 ((ba ) ∩ b) = (b ∩ (ba ))
54bi1 118 . . . . . . . 8 (((ba ) ∩ b) ≡ (b ∩ (ba ))) = 1
63, 5wr2 371 . . . . . . 7 (((ab) ∩ b) ≡ (b ∩ (ba ))) = 1
7 anabs 121 . . . . . . . 8 (b ∩ (ba )) = b
87bi1 118 . . . . . . 7 ((b ∩ (ba )) ≡ b) = 1
96, 8wr2 371 . . . . . 6 (((ab) ∩ b) ≡ b) = 1
109wlan 370 . . . . 5 (((ab ) ∩ ((ab) ∩ b)) ≡ ((ab ) ∩ b)) = 1
11 wcomlem.1 . . . . . . . . . 10 (a ≡ ((ab) ∪ (ab ))) = 1
12 df-a 40 . . . . . . . . . . . 12 (ab) = (ab )
1312bi1 118 . . . . . . . . . . 11 ((ab) ≡ (ab ) ) = 1
14 anor1 88 . . . . . . . . . . . 12 (ab ) = (ab)
1514bi1 118 . . . . . . . . . . 11 ((ab ) ≡ (ab) ) = 1
1613, 15w2or 372 . . . . . . . . . 10 (((ab) ∪ (ab )) ≡ ((ab ) ∪ (ab) )) = 1
1711, 16wr2 371 . . . . . . . . 9 (a ≡ ((ab ) ∪ (ab) )) = 1
1817wr4 199 . . . . . . . 8 (a ≡ ((ab ) ∪ (ab) ) ) = 1
19 df-a 40 . . . . . . . . . 10 ((ab ) ∩ (ab)) = ((ab ) ∪ (ab) )
2019bi1 118 . . . . . . . . 9 (((ab ) ∩ (ab)) ≡ ((ab ) ∪ (ab) ) ) = 1
2120wr1 197 . . . . . . . 8 (((ab ) ∪ (ab) ) ≡ ((ab ) ∩ (ab))) = 1
2218, 21wr2 371 . . . . . . 7 (a ≡ ((ab ) ∩ (ab))) = 1
2322wran 369 . . . . . 6 ((ab) ≡ (((ab ) ∩ (ab)) ∩ b)) = 1
24 anass 76 . . . . . . 7 (((ab ) ∩ (ab)) ∩ b) = ((ab ) ∩ ((ab) ∩ b))
2524bi1 118 . . . . . 6 ((((ab ) ∩ (ab)) ∩ b) ≡ ((ab ) ∩ ((ab) ∩ b))) = 1
2623, 25wr2 371 . . . . 5 ((ab) ≡ ((ab ) ∩ ((ab) ∩ b))) = 1
2713wcon2 208 . . . . . 6 ((ab) ≡ (ab )) = 1
2827wran 369 . . . . 5 (((ab)b) ≡ ((ab ) ∩ b)) = 1
2910, 26, 28w3tr1 374 . . . 4 ((ab) ≡ ((ab)b)) = 1
3029wlor 368 . . 3 (((ab) ∪ (ab)) ≡ ((ab) ∪ ((ab)b))) = 1
3130wr1 197 . 2 (((ab) ∪ ((ab)b)) ≡ ((ab) ∪ (ab))) = 1
32 ax-a2 31 . . . . . 6 ((ab) ∪ b) = (b ∪ (ab))
3332bi1 118 . . . . 5 (((ab) ∪ b) ≡ (b ∪ (ab))) = 1
34 ancom 74 . . . . . . . 8 (ab) = (ba)
3534bi1 118 . . . . . . 7 ((ab) ≡ (ba)) = 1
3635wlor 368 . . . . . 6 ((b ∪ (ab)) ≡ (b ∪ (ba))) = 1
37 orabs 120 . . . . . . 7 (b ∪ (ba)) = b
3837bi1 118 . . . . . 6 ((b ∪ (ba)) ≡ b) = 1
3936, 38wr2 371 . . . . 5 ((b ∪ (ab)) ≡ b) = 1
4033, 39wr2 371 . . . 4 (((ab) ∪ b) ≡ b) = 1
4140wdf-le1 378 . . 3 ((ab) ≤2 b) = 1
4241wom4 380 . 2 (((ab) ∪ ((ab)b)) ≡ b) = 1
43 ancom 74 . . . 4 (ab) = (ba )
4443bi1 118 . . 3 ((ab) ≡ (ba )) = 1
4535, 44w2or 372 . 2 (((ab) ∪ (ab)) ≡ ((ba) ∪ (ba ))) = 1
4631, 42, 45w3tr2 375 1 (b ≡ ((ba) ∪ (ba ))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8 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 This theorem is referenced by:  wdf-c1  383
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