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Theorem mlaoml 833
Description: Mladen's OML. (Contributed by NM, 4-Nov-1998.)
Assertion
Ref Expression
mlaoml ((ab) ∩ (bc)) ≤ (ac)

Proof of Theorem mlaoml
StepHypRef Expression
1 u1lembi 720 . . . . 5 ((a1 b) ∩ (b1 a)) = (ab)
21ran 78 . . . 4 (((a1 b) ∩ (b1 a)) ∩ (b1 c)) = ((ab) ∩ (b1 c))
3 mlalem 832 . . . 4 ((ab) ∩ (b1 c)) ≤ (a1 c)
42, 3bltr 138 . . 3 (((a1 b) ∩ (b1 a)) ∩ (b1 c)) ≤ (a1 c)
5 ancom 74 . . . . . 6 ((b1 a) ∩ (c1 b)) = ((c1 b) ∩ (b1 a))
65ran 78 . . . . 5 (((b1 a) ∩ (c1 b)) ∩ (b1 c)) = (((c1 b) ∩ (b1 a)) ∩ (b1 c))
7 an32 83 . . . . 5 (((c1 b) ∩ (b1 a)) ∩ (b1 c)) = (((c1 b) ∩ (b1 c)) ∩ (b1 a))
8 u1lembi 720 . . . . . 6 ((c1 b) ∩ (b1 c)) = (cb)
98ran 78 . . . . 5 (((c1 b) ∩ (b1 c)) ∩ (b1 a)) = ((cb) ∩ (b1 a))
106, 7, 93tr 65 . . . 4 (((b1 a) ∩ (c1 b)) ∩ (b1 c)) = ((cb) ∩ (b1 a))
11 mlalem 832 . . . 4 ((cb) ∩ (b1 a)) ≤ (c1 a)
1210, 11bltr 138 . . 3 (((b1 a) ∩ (c1 b)) ∩ (b1 c)) ≤ (c1 a)
134, 12le2an 169 . 2 ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (((b1 a) ∩ (c1 b)) ∩ (b1 c))) ≤ ((a1 c) ∩ (c1 a))
14 an12 81 . . . . . 6 ((b1 a) ∩ ((a1 b) ∩ (c1 b))) = ((a1 b) ∩ ((b1 a) ∩ (c1 b)))
15 ancom 74 . . . . . . . 8 ((a1 b) ∩ (b1 a)) = ((b1 a) ∩ (a1 b))
1615ran 78 . . . . . . 7 (((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b))) = (((b1 a) ∩ (a1 b)) ∩ ((b1 a) ∩ (c1 b)))
17 id 59 . . . . . . 7 (((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b))) = (((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b)))
18 anandi 114 . . . . . . 7 ((b1 a) ∩ ((a1 b) ∩ (c1 b))) = (((b1 a) ∩ (a1 b)) ∩ ((b1 a) ∩ (c1 b)))
1916, 17, 183tr1 63 . . . . . 6 (((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b))) = ((b1 a) ∩ ((a1 b) ∩ (c1 b)))
20 anass 76 . . . . . 6 (((a1 b) ∩ (b1 a)) ∩ (c1 b)) = ((a1 b) ∩ ((b1 a) ∩ (c1 b)))
2114, 19, 203tr1 63 . . . . 5 (((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b))) = (((a1 b) ∩ (b1 a)) ∩ (c1 b))
2221ran 78 . . . 4 ((((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b))) ∩ (b1 c)) = ((((a1 b) ∩ (b1 a)) ∩ (c1 b)) ∩ (b1 c))
23 anandir 115 . . . 4 ((((a1 b) ∩ (b1 a)) ∩ ((b1 a) ∩ (c1 b))) ∩ (b1 c)) = ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (((b1 a) ∩ (c1 b)) ∩ (b1 c)))
24 an32 83 . . . 4 ((((a1 b) ∩ (b1 a)) ∩ (c1 b)) ∩ (b1 c)) = ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (c1 b))
2522, 23, 243tr2 64 . . 3 ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (((b1 a) ∩ (c1 b)) ∩ (b1 c))) = ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (c1 b))
26 anass 76 . . 3 ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (c1 b)) = (((a1 b) ∩ (b1 a)) ∩ ((b1 c) ∩ (c1 b)))
27 u1lembi 720 . . . 4 ((b1 c) ∩ (c1 b)) = (bc)
281, 272an 79 . . 3 (((a1 b) ∩ (b1 a)) ∩ ((b1 c) ∩ (c1 b))) = ((ab) ∩ (bc))
2925, 26, 283tr 65 . 2 ((((a1 b) ∩ (b1 a)) ∩ (b1 c)) ∩ (((b1 a) ∩ (c1 b)) ∩ (b1 c))) = ((ab) ∩ (bc))
30 u1lembi 720 . 2 ((a1 c) ∩ (c1 a)) = (ac)
3113, 29, 30le3tr2 141 1 ((ab) ∩ (bc)) ≤ (ac)
Colors of variables: term
Syntax hints:  wle 2  tb 5  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:  eqtr4  834  mlaconj4  844
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