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Theorem gomaex3lem9 922
 Description: Lemma for Godowski 6-var -> Mayet Example 3. (Contributed by NM, 29-Nov-1999.)
Hypotheses
Ref Expression
gomaex3lem5.1 ab
gomaex3lem5.2 bc
gomaex3lem5.3 cd
gomaex3lem5.5 ef
gomaex3lem5.6 fa
gomaex3lem5.8 (((i2 g) ∩ (g2 y)) ∩ (((y2 w) ∩ (w2 n)) ∩ ((n2 k) ∩ (k2 i)))) ≤ (g2 i)
gomaex3lem5.9 p = ((ab) →1 (de) )
gomaex3lem5.10 q = ((ef) →1 (bc) )
gomaex3lem5.11 r = ((p1 q) ∩ (cd))
gomaex3lem5.12 g = a
gomaex3lem5.13 h = b
gomaex3lem5.14 i = c
gomaex3lem5.15 j = (cd)
gomaex3lem5.16 k = r
gomaex3lem5.17 m = (p1 q)
gomaex3lem5.18 n = (p1 q)
gomaex3lem5.19 u = (pq)
gomaex3lem5.20 w = q
gomaex3lem5.21 x = q
gomaex3lem5.22 y = (ef)
gomaex3lem5.23 z = f
Assertion
Ref Expression
gomaex3lem9 (((ab) ∩ (de) ) ∩ (r ∪ (p1 q))) ≤ (bc)

Proof of Theorem gomaex3lem9
StepHypRef Expression
1 ancom 74 . . 3 (((ab) ∩ (de) ) ∩ (r ∪ (p1 q))) = ((r ∪ (p1 q)) ∩ ((ab) ∩ (de) ))
2 gomaex3lem5.9 . . . . . . 7 p = ((ab) →1 (de) )
32gomaex3lem4 917 . . . . . 6 ((ab) ∩ (de) ) ≤ p
43df2le2 136 . . . . 5 (((ab) ∩ (de) ) ∩ p ) = ((ab) ∩ (de) )
54ax-r1 35 . . . 4 ((ab) ∩ (de) ) = (((ab) ∩ (de) ) ∩ p )
65lan 77 . . 3 ((r ∪ (p1 q)) ∩ ((ab) ∩ (de) )) = ((r ∪ (p1 q)) ∩ (((ab) ∩ (de) ) ∩ p ))
7 an12 81 . . 3 ((r ∪ (p1 q)) ∩ (((ab) ∩ (de) ) ∩ p )) = (((ab) ∩ (de) ) ∩ ((r ∪ (p1 q)) ∩ p ))
81, 6, 73tr 65 . 2 (((ab) ∩ (de) ) ∩ (r ∪ (p1 q))) = (((ab) ∩ (de) ) ∩ ((r ∪ (p1 q)) ∩ p ))
9 gomaex3lem5.1 . . 3 ab
10 gomaex3lem5.2 . . 3 bc
11 gomaex3lem5.3 . . 3 cd
12 gomaex3lem5.5 . . 3 ef
13 gomaex3lem5.6 . . 3 fa
14 gomaex3lem5.8 . . 3 (((i2 g) ∩ (g2 y)) ∩ (((y2 w) ∩ (w2 n)) ∩ ((n2 k) ∩ (k2 i)))) ≤ (g2 i)
15 gomaex3lem5.10 . . 3 q = ((ef) →1 (bc) )
16 gomaex3lem5.11 . . 3 r = ((p1 q) ∩ (cd))
17 gomaex3lem5.12 . . 3 g = a
18 gomaex3lem5.13 . . 3 h = b
19 gomaex3lem5.14 . . 3 i = c
20 gomaex3lem5.15 . . 3 j = (cd)
21 gomaex3lem5.16 . . 3 k = r
22 gomaex3lem5.17 . . 3 m = (p1 q)
23 gomaex3lem5.18 . . 3 n = (p1 q)
24 gomaex3lem5.19 . . 3 u = (pq)
25 gomaex3lem5.20 . . 3 w = q
26 gomaex3lem5.21 . . 3 x = q
27 gomaex3lem5.22 . . 3 y = (ef)
28 gomaex3lem5.23 . . 3 z = f
299, 10, 11, 12, 13, 14, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28gomaex3lem8 921 . 2 (((ab) ∩ (de) ) ∩ ((r ∪ (p1 q)) ∩ p )) ≤ (bc)
308, 29bltr 138 1 (((ab) ∩ (de) ) ∩ (r ∪ (p1 q))) ≤ (bc)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  gomaex3lem10  923
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