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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' =< (a ->1 c)
Assertion
Ref Expression
kb10iii c' =< (a ->1 b)
Proof of Theorem kb10iii
StepHypRef Expression
1 ud1lem0c 277 . . 3 (a ->1 b)' = (a ^ (a' v b'))
2 omln 446 . . . . . . . 8 (a' v (a ^ (a' v b'))) = (a' v b')
3 u1lem9b 778 . . . . . . . . 9 a' =< (a ->1 c)
4 kb10iii.1 . . . . . . . . 9 b' =< (a ->1 c)
53, 4lel2or 170 . . . . . . . 8 (a' v b') =< (a ->1 c)
62, 5bltr 138 . . . . . . 7 (a' v (a ^ (a' v b'))) =< (a ->1 c)
76lelan 167 . . . . . 6 (a ^ (a' v (a ^ (a' v b')))) =< (a ^ (a ->1 c))
8 ancom 74 . . . . . 6 (a ^ (a ->1 c)) = ((a ->1 c) ^ a)
97, 8lbtr 139 . . . . 5 (a ^ (a' v (a ^ (a' v b')))) =< ((a ->1 c) ^ a)
10 womaon 221 . . . . 5 (a ^ (a' v (a ^ (a' v b')))) = (a ^ (a' v b'))
11 u1lemaa 600 . . . . 5 ((a ->1 c) ^ a) = (a ^ c)
129, 10, 11le3tr2 141 . . . 4 (a ^ (a' v b')) =< (a ^ c)
13 lear 161 . . . 4 (a ^ c) =< c
1412, 13letr 137 . . 3 (a ^ (a' v b')) =< c
151, 14bltr 138 . 2 (a ->1 b)' =< c
1615lecon2 156 1 c' =< (a ->1 b)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v 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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