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Theorem oas 925
Description: "Strengthening" lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oas.1 (a' ^ (a v b)) =< c
Assertion
Ref Expression
oas ((a ->1 c) ^ (a v b)) =< c
Proof of Theorem oas
StepHypRef Expression
1 oml 445 . . . . . 6 (a v (a' ^ (a v b))) = (a v b)
21ax-r1 35 . . . . 5 (a v b) = (a v (a' ^ (a v b)))
3 lea 160 . . . . . . 7 (a' ^ (a v b)) =< a'
4 oas.1 . . . . . . 7 (a' ^ (a v b)) =< c
53, 4ler2an 173 . . . . . 6 (a' ^ (a v b)) =< (a' ^ c)
65lelor 166 . . . . 5 (a v (a' ^ (a v b))) =< (a v (a' ^ c))
72, 6bltr 138 . . . 4 (a v b) =< (a v (a' ^ c))
87lelan 167 . . 3 ((a ->1 c) ^ (a v b)) =< ((a ->1 c) ^ (a v (a' ^ c)))
9 u1lemc1 680 . . . . 5 a C (a ->1 c)
10 comanr1 464 . . . . . 6 a' C (a' ^ c)
1110comcom6 459 . . . . 5 a C (a' ^ c)
129, 11fh2 470 . . . 4 ((a ->1 c) ^ (a v (a' ^ c))) = (((a ->1 c) ^ a) v ((a ->1 c) ^ (a' ^ c)))
13 u1lemaa 600 . . . . 5 ((a ->1 c) ^ a) = (a ^ c)
14 ancom 74 . . . . . 6 ((a ->1 c) ^ (a' ^ c)) = ((a' ^ c) ^ (a ->1 c))
15 lea 160 . . . . . . . 8 (a' ^ c) =< a'
16 leo 158 . . . . . . . . 9 a' =< (a' v (a ^ c))
17 df-i1 44 . . . . . . . . . 10 (a ->1 c) = (a' v (a ^ c))
1817ax-r1 35 . . . . . . . . 9 (a' v (a ^ c)) = (a ->1 c)
1916, 18lbtr 139 . . . . . . . 8 a' =< (a ->1 c)
2015, 19letr 137 . . . . . . 7 (a' ^ c) =< (a ->1 c)
2120df2le2 136 . . . . . 6 ((a' ^ c) ^ (a ->1 c)) = (a' ^ c)
2214, 21ax-r2 36 . . . . 5 ((a ->1 c) ^ (a' ^ c)) = (a' ^ c)
2313, 222or 72 . . . 4 (((a ->1 c) ^ a) v ((a ->1 c) ^ (a' ^ c))) = ((a ^ c) v (a' ^ c))
2412, 23ax-r2 36 . . 3 ((a ->1 c) ^ (a v (a' ^ c))) = ((a ^ c) v (a' ^ c))
258, 24lbtr 139 . 2 ((a ->1 c) ^ (a v b)) =< ((a ^ c) v (a' ^ c))
26 lear 161 . . 3 (a ^ c) =< c
27 lear 161 . . 3 (a' ^ c) =< c
2826, 27lel2or 170 . 2 ((a ^ c) v (a' ^ c)) =< c
2925, 28letr 137 1 ((a ->1 c) ^ (a v b)) =< c
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:  oa4ctob  967  oa3-2wto2  989
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