Theorem kb10iii 875

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

Step	Hyp	Ref	Expression
1		ud1lem0c 269	. . 3 (a ->1 b)_|_ = (a ^ (a_|_ v b_|_))
2		omln 428	. . . . . . . 8 (a_|_ v (a ^ (a_|_ v b_|_))) = (a_|_ v b_|_)
3		u1lem9b 760	. . . . . . . . 9 a_|_ =< (a ->1 c)
4		kb10iii.1	. . . . . . . . 9 b_|_ =< (a ->1 c)
5	3, 4	lel2or 162	. . . . . . . 8 (a_|_ v b_|_) =< (a ->1 c)
6	2, 5	bltr 130	. . . . . . 7 (a_|_ v (a ^ (a_|_ v b_|_))) =< (a ->1 c)
7	6	lelan 159	. . . . . 6 (a ^ (a_|_ v (a ^ (a_|_ v b_|_)))) =< (a ^ (a ->1 c))
8		ancom 68	. . . . . 6 (a ^ (a ->1 c)) = ((a ->1 c) ^ a)
9	7, 8	lbtr 131	. . . . 5 (a ^ (a_|_ v (a ^ (a_|_ v b_|_)))) =< ((a ->1 c) ^ a)
10		womaon 213	. . . . 5 (a ^ (a_|_ v (a ^ (a_|_ v b_|_)))) = (a ^ (a_|_ v b_|_))
11		u1lemaa 582	. . . . 5 ((a ->1 c) ^ a) = (a ^ c)
12	9, 10, 11	le3tr2 133	. . . 4 (a ^ (a_|_ v b_|_)) =< (a ^ c)
13		lear 153	. . . 4 (a ^ c) =< c
14	12, 13	letr 129	. . 3 (a ^ (a_|_ v b_|_)) =< c
15	1, 14	bltr 130	. 2 (a ->1 b)_|_ =< c
16	15	lecon2 148	1 c_|_ =< (a ->1 b)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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