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Theorem go1 343
 Description: Lemma for proof of Mayet 8-variable "full" equation from 4-variable Godowski equation.
Assertion
Ref Expression
go1 ((ab) ∩ (a1 b )) = 0

Proof of Theorem go1
StepHypRef Expression
1 df-i1 44 . . 3 (a1 b ) = (a ∪ (ab ))
21lan 77 . 2 ((ab) ∩ (a1 b )) = ((ab) ∩ (a ∪ (ab )))
3 lear 161 . . . . . 6 (ab ) ≤ b
43lelor 166 . . . . 5 (a ∪ (ab )) ≤ (ab )
54lelan 167 . . . 4 ((ab) ∩ (a ∪ (ab ))) ≤ ((ab) ∩ (ab ))
6 oran3 93 . . . . . 6 (ab ) = (ab)
76lan 77 . . . . 5 ((ab) ∩ (ab )) = ((ab) ∩ (ab) )
8 dff 101 . . . . . 6 0 = ((ab) ∩ (ab) )
98ax-r1 35 . . . . 5 ((ab) ∩ (ab) ) = 0
107, 9ax-r2 36 . . . 4 ((ab) ∩ (ab )) = 0
115, 10lbtr 139 . . 3 ((ab) ∩ (a ∪ (ab ))) ≤ 0
12 le0 147 . . 3 0 ≤ ((ab) ∩ (a ∪ (ab )))
1311, 12lebi 145 . 2 ((ab) ∩ (a ∪ (ab ))) = 0
142, 13ax-r2 36 1 ((ab) ∩ (a1 b )) = 0
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131 This theorem is referenced by:  gomaex4  900
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