Theorem u1lem9a 777

Description: Lemma used in study of orthoarguesian law. Equation 4.11 of [MegPav2000] p. 23. This is the first part of the inequality.

Assertion

Ref	Expression
u1lem9a	(a' ->1 b)' =< a'

Proof of Theorem u1lem9a

Step	Hyp	Ref	Expression
1		df-i1 44	. . . 4 (a' ->1 b) = (a'' v (a' ^ b))
2	1	ax-r4 37	. . 3 (a' ->1 b)' = (a'' v (a' ^ b))'
3		anor1 88	. . . 4 (a' ^ (a' ^ b)') = (a'' v (a' ^ b))'
4	3	ax-r1 35	. . 3 (a'' v (a' ^ b))' = (a' ^ (a' ^ b)')
5	2, 4	ax-r2 36	. 2 (a' ->1 b)' = (a' ^ (a' ^ b)')
6		lea 160	. 2 (a' ^ (a' ^ b)') =< a'
7	5, 6	bltr 138	1 (a' ->1 b)' =< a'

Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12

This theorem is referenced by:  u1lem9ab 779  sadm3 838  oa4uto4g 975  oa4uto4 977  lem4.6.3le1 1081

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i1 44  df-le1 130  df-le2 131

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