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Theorem wcom3ii 419
 Description: Lemma 3(ii) of Kalmbach 83 p. 23. (Contributed by NM, 13-Oct-1997.)
Hypothesis
Ref Expression
wcomcom.1 C (a, b) = 1
Assertion
Ref Expression
wcom3ii ((a ∩ (ab)) ≡ (ab)) = 1

Proof of Theorem wcom3ii
StepHypRef Expression
1 wcomcom.1 . . . . . 6 C (a, b) = 1
21wcomcom 414 . . . . 5 C (b, a) = 1
32wcomd 418 . . . 4 (b ≡ ((ba) ∩ (ba ))) = 1
43wlan 370 . . 3 ((ab) ≡ (a ∩ ((ba) ∩ (ba )))) = 1
5 anass 76 . . . . . 6 ((a ∩ (ba)) ∩ (ba )) = (a ∩ ((ba) ∩ (ba )))
65bi1 118 . . . . 5 (((a ∩ (ba)) ∩ (ba )) ≡ (a ∩ ((ba) ∩ (ba )))) = 1
76wr1 197 . . . 4 ((a ∩ ((ba) ∩ (ba ))) ≡ ((a ∩ (ba)) ∩ (ba ))) = 1
8 ax-a2 31 . . . . . . . 8 (ba) = (ab)
98bi1 118 . . . . . . 7 ((ba) ≡ (ab)) = 1
109wlan 370 . . . . . 6 ((a ∩ (ba)) ≡ (a ∩ (ab))) = 1
11 anabs 121 . . . . . . 7 (a ∩ (ab)) = a
1211bi1 118 . . . . . 6 ((a ∩ (ab)) ≡ a) = 1
1310, 12wr2 371 . . . . 5 ((a ∩ (ba)) ≡ a) = 1
14 ax-a2 31 . . . . . 6 (ba ) = (ab)
1514bi1 118 . . . . 5 ((ba ) ≡ (ab)) = 1
1613, 15w2an 373 . . . 4 (((a ∩ (ba)) ∩ (ba )) ≡ (a ∩ (ab))) = 1
177, 16wr2 371 . . 3 ((a ∩ ((ba) ∩ (ba ))) ≡ (a ∩ (ab))) = 1
184, 17wr2 371 . 2 ((ab) ≡ (a ∩ (ab))) = 1
1918wr1 197 1 ((a ∩ (ab)) ≡ (ab)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29 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:  wfh1  423  wfh2  424
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