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Theorem u1lem8 776
Description: Lemma used in study of orthoarguesian law.
Assertion
Ref Expression
u1lem8 ((a ->1 b) ^ (a' ->1 b)) = ((a ^ b) v (a' ^ b))
Proof of Theorem u1lem8
StepHypRef Expression
1 df-i1 44 . . 3 (a ->1 b) = (a' v (a ^ b))
2 df-i1 44 . . . 4 (a' ->1 b) = (a'' v (a' ^ b))
3 ax-a1 30 . . . . . 6 a = a''
43ax-r5 38 . . . . 5 (a v (a' ^ b)) = (a'' v (a' ^ b))
54ax-r1 35 . . . 4 (a'' v (a' ^ b)) = (a v (a' ^ b))
62, 5ax-r2 36 . . 3 (a' ->1 b) = (a v (a' ^ b))
71, 62an 79 . 2 ((a ->1 b) ^ (a' ->1 b)) = ((a' v (a ^ b)) ^ (a v (a' ^ b)))
8 comor1 461 . . . 4 (a v (a' ^ b)) C a
98comcom2 183 . . 3 (a v (a' ^ b)) C a'
10 coman1 185 . . . . 5 (a ^ b) C a
1110comcom2 183 . . . . . 6 (a ^ b) C a'
12 coman2 186 . . . . . 6 (a ^ b) C b
1311, 12com2an 484 . . . . 5 (a ^ b) C (a' ^ b)
1410, 13com2or 483 . . . 4 (a ^ b) C (a v (a' ^ b))
1514comcom 453 . . 3 (a v (a' ^ b)) C (a ^ b)
169, 15fh1r 473 . 2 ((a' v (a ^ b)) ^ (a v (a' ^ b))) = ((a' ^ (a v (a' ^ b))) v ((a ^ b) ^ (a v (a' ^ b))))
17 omlan 448 . . . 4 (a' ^ (a v (a' ^ b))) = (a' ^ b)
18 lea 160 . . . . . 6 (a ^ b) =< a
19 leo 158 . . . . . 6 a =< (a v (a' ^ b))
2018, 19letr 137 . . . . 5 (a ^ b) =< (a v (a' ^ b))
2120df2le2 136 . . . 4 ((a ^ b) ^ (a v (a' ^ b))) = (a ^ b)
2217, 212or 72 . . 3 ((a' ^ (a v (a' ^ b))) v ((a ^ b) ^ (a v (a' ^ b)))) = ((a' ^ b) v (a ^ b))
23 ax-a2 31 . . 3 ((a' ^ b) v (a ^ b)) = ((a ^ b) v (a' ^ b))
2422, 23ax-r2 36 . 2 ((a' ^ (a v (a' ^ b))) v ((a ^ b) ^ (a v (a' ^ b)))) = ((a ^ b) v (a' ^ b))
257, 16, 243tr 65 1 ((a ->1 b) ^ (a' ->1 b)) = ((a ^ b) v (a' ^ b))
Colors of variables: term
Syntax hints:   = wb 1  '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:  oa4to4u  973  oa4to4u2  974  oa3-u1  991  oa3-u2  992
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