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Theorem kb10iii 893
 Description: Exercise 10(iii) of Kalmbach p. 30 (in a rewritten form).
Hypothesis
Ref Expression
kb10iii.1 b ≤ (a1 c)
Assertion
Ref Expression
kb10iii c ≤ (a1 b)

Proof of Theorem kb10iii
StepHypRef Expression
1 ud1lem0c 277 . . 3 (a1 b) = (a ∩ (ab ))
2 omln 446 . . . . . . . 8 (a ∪ (a ∩ (ab ))) = (ab )
3 u1lem9b 778 . . . . . . . . 9 a ≤ (a1 c)
4 kb10iii.1 . . . . . . . . 9 b ≤ (a1 c)
53, 4lel2or 170 . . . . . . . 8 (ab ) ≤ (a1 c)
62, 5bltr 138 . . . . . . 7 (a ∪ (a ∩ (ab ))) ≤ (a1 c)
76lelan 167 . . . . . 6 (a ∩ (a ∪ (a ∩ (ab )))) ≤ (a ∩ (a1 c))
8 ancom 74 . . . . . 6 (a ∩ (a1 c)) = ((a1 c) ∩ a)
97, 8lbtr 139 . . . . 5 (a ∩ (a ∪ (a ∩ (ab )))) ≤ ((a1 c) ∩ a)
10 womaon 221 . . . . 5 (a ∩ (a ∪ (a ∩ (ab )))) = (a ∩ (ab ))
11 u1lemaa 600 . . . . 5 ((a1 c) ∩ a) = (ac)
129, 10, 11le3tr2 141 . . . 4 (a ∩ (ab )) ≤ (ac)
13 lear 161 . . . 4 (ac) ≤ c
1412, 13letr 137 . . 3 (a ∩ (ab )) ≤ c
151, 14bltr 138 . 2 (a1 b)c
1615lecon2 156 1 c ≤ (a1 b)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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