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Theorem gomaex3lem7 920
 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
gomaex3lem7 (((ab) ∩ d ) ∩ (((r ∪ (p1 q)) ∩ p ) ∩ e )) ≤ (bc)

Proof of Theorem gomaex3lem7
StepHypRef Expression
1 gomaex3lem5.3 . . . . . 6 cd
21gomaex3lem1 914 . . . . 5 (c ∪ (cd) ) = d
32lan 77 . . . 4 ((ab) ∩ (c ∪ (cd) )) = ((ab) ∩ d )
4 gomaex3lem3 916 . . . . . 6 ((p1 q) ∪ (pq)) = p
54lan 77 . . . . 5 ((r ∪ (p1 q)) ∩ ((p1 q) ∪ (pq))) = ((r ∪ (p1 q)) ∩ p )
6 ancom 74 . . . . . 6 ((qq) ∩ ((ef)f)) = (((ef)f) ∩ (qq))
7 gomaex3lem5.5 . . . . . . . 8 ef
87gomaex3lem2 915 . . . . . . 7 ((ef)f) = e
9 ax-a2 31 . . . . . . . 8 (qq) = (qq )
10 df-t 41 . . . . . . . . 9 1 = (qq )
1110ax-r1 35 . . . . . . . 8 (qq ) = 1
129, 11ax-r2 36 . . . . . . 7 (qq) = 1
138, 122an 79 . . . . . 6 (((ef)f) ∩ (qq)) = (e ∩ 1)
14 an1 106 . . . . . 6 (e ∩ 1) = e
156, 13, 143tr 65 . . . . 5 ((qq) ∩ ((ef)f)) = e
165, 152an 79 . . . 4 (((r ∪ (p1 q)) ∩ ((p1 q) ∪ (pq))) ∩ ((qq) ∩ ((ef)f))) = (((r ∪ (p1 q)) ∩ p ) ∩ e )
173, 162an 79 . . 3 (((ab) ∩ (c ∪ (cd) )) ∩ (((r ∪ (p1 q)) ∩ ((p1 q) ∪ (pq))) ∩ ((qq) ∩ ((ef)f)))) = (((ab) ∩ d ) ∩ (((r ∪ (p1 q)) ∩ p ) ∩ e ))
1817ax-r1 35 . 2 (((ab) ∩ d ) ∩ (((r ∪ (p1 q)) ∩ p ) ∩ e )) = (((ab) ∩ (c ∪ (cd) )) ∩ (((r ∪ (p1 q)) ∩ ((p1 q) ∪ (pq))) ∩ ((qq) ∩ ((ef)f))))
19 gomaex3lem5.1 . . 3 ab
20 gomaex3lem5.2 . . 3 bc
21 gomaex3lem5.6 . . 3 fa
22 gomaex3lem5.8 . . 3 (((i2 g) ∩ (g2 y)) ∩ (((y2 w) ∩ (w2 n)) ∩ ((n2 k) ∩ (k2 i)))) ≤ (g2 i)
23 gomaex3lem5.9 . . 3 p = ((ab) →1 (de) )
24 gomaex3lem5.10 . . 3 q = ((ef) →1 (bc) )
25 gomaex3lem5.11 . . 3 r = ((p1 q) ∩ (cd))
26 gomaex3lem5.12 . . 3 g = a
27 gomaex3lem5.13 . . 3 h = b
28 gomaex3lem5.14 . . 3 i = c
29 gomaex3lem5.15 . . 3 j = (cd)
30 gomaex3lem5.16 . . 3 k = r
31 gomaex3lem5.17 . . 3 m = (p1 q)
32 gomaex3lem5.18 . . 3 n = (p1 q)
33 gomaex3lem5.19 . . 3 u = (pq)
34 gomaex3lem5.20 . . 3 w = q
35 gomaex3lem5.21 . . 3 x = q
36 gomaex3lem5.22 . . 3 y = (ef)
37 gomaex3lem5.23 . . 3 z = f
3819, 20, 1, 7, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37gomaex3lem6 919 . 2 (((ab) ∩ (c ∪ (cd) )) ∩ (((r ∪ (p1 q)) ∩ ((p1 q) ∪ (pq))) ∩ ((qq) ∩ ((ef)f)))) ≤ (bc)
3918, 38bltr 138 1 (((ab) ∩ d ) ∩ (((r ∪ (p1 q)) ∩ p ) ∩ e )) ≤ (bc)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  gomaex3lem8  921
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