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Theorem u1lem8 776
 Description: Lemma used in study of orthoarguesian law.
Assertion
Ref Expression
u1lem8 ((a1 b) ∩ (a1 b)) = ((ab) ∪ (ab))

Proof of Theorem u1lem8
StepHypRef Expression
1 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
2 df-i1 44 . . . 4 (a1 b) = (a ∪ (ab))
3 ax-a1 30 . . . . . 6 a = a
43ax-r5 38 . . . . 5 (a ∪ (ab)) = (a ∪ (ab))
54ax-r1 35 . . . 4 (a ∪ (ab)) = (a ∪ (ab))
62, 5ax-r2 36 . . 3 (a1 b) = (a ∪ (ab))
71, 62an 79 . 2 ((a1 b) ∩ (a1 b)) = ((a ∪ (ab)) ∩ (a ∪ (ab)))
8 comor1 461 . . . 4 (a ∪ (ab)) C a
98comcom2 183 . . 3 (a ∪ (ab)) C a
10 coman1 185 . . . . 5 (ab) C a
1110comcom2 183 . . . . . 6 (ab) C a
12 coman2 186 . . . . . 6 (ab) C b
1311, 12com2an 484 . . . . 5 (ab) C (ab)
1410, 13com2or 483 . . . 4 (ab) C (a ∪ (ab))
1514comcom 453 . . 3 (a ∪ (ab)) C (ab)
169, 15fh1r 473 . 2 ((a ∪ (ab)) ∩ (a ∪ (ab))) = ((a ∩ (a ∪ (ab))) ∪ ((ab) ∩ (a ∪ (ab))))
17 omlan 448 . . . 4 (a ∩ (a ∪ (ab))) = (ab)
18 lea 160 . . . . . 6 (ab) ≤ a
19 leo 158 . . . . . 6 a ≤ (a ∪ (ab))
2018, 19letr 137 . . . . 5 (ab) ≤ (a ∪ (ab))
2120df2le2 136 . . . 4 ((ab) ∩ (a ∪ (ab))) = (ab)
2217, 212or 72 . . 3 ((a ∩ (a ∪ (ab))) ∪ ((ab) ∩ (a ∪ (ab)))) = ((ab) ∪ (ab))
23 ax-a2 31 . . 3 ((ab) ∪ (ab)) = ((ab) ∪ (ab))
2422, 23ax-r2 36 . 2 ((a ∩ (a ∪ (ab))) ∪ ((ab) ∩ (a ∪ (ab)))) = ((ab) ∪ (ab))
257, 16, 243tr 65 1 ((a1 b) ∩ (a1 b)) = ((ab) ∪ (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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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