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Theorem gomaex3lem2 915
 Description: Lemma for Godowski 6-var -> Mayet Example 3.
Hypothesis
Ref Expression
gomaex3lem2.5 ef
Assertion
Ref Expression
gomaex3lem2 ((ef)f) = e

Proof of Theorem gomaex3lem2
StepHypRef Expression
1 gomaex3lem2.5 . . . . . 6 ef
21lecon3 157 . . . . 5 fe
32lecom 180 . . . 4 f C e
4 comid 187 . . . . 5 f C f
54comcom2 183 . . . 4 f C f
63, 5fh3r 475 . . 3 ((ef ) ∪ f) = ((ef) ∩ (ff))
7 anor3 90 . . . . 5 (ef ) = (ef)
87ax-r5 38 . . . 4 ((ef ) ∪ f) = ((ef)f)
98ax-r1 35 . . 3 ((ef)f) = ((ef ) ∪ f)
10 anabs 121 . . . . . 6 (e ∩ (ef)) = e
1110df2le1 135 . . . . 5 e ≤ (ef)
12 leid 148 . . . . . 6 ee
1312, 2lel2or 170 . . . . 5 (ef) ≤ e
1411, 13lebi 145 . . . 4 e = (ef)
15 df-t 41 . . . . 5 1 = (ff )
16 ax-a2 31 . . . . 5 (ff ) = (ff)
1715, 16ax-r2 36 . . . 4 1 = (ff)
1814, 172an 79 . . 3 (e ∩ 1) = ((ef) ∩ (ff))
196, 9, 183tr1 63 . 2 ((ef)f) = (e ∩ 1)
20 an1 106 . 2 (e ∩ 1) = e
2119, 20ax-r2 36 1 ((ef)f) = e
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ 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-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:  gomaex3lem7  920
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