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Theorem mhlem1 877
 Description: Lemma for Marsden-Herman distributive law.
Hypotheses
Ref Expression
mhlem1.1 a C b
mhlem1.2 c C b
Assertion
Ref Expression
mhlem1 ((ab) ∩ (bc)) = ((ab ) ∪ (bc))

Proof of Theorem mhlem1
StepHypRef Expression
1 df-t 41 . . . . 5 1 = (bb )
21lan 77 . . . 4 ((ab) ∩ 1) = ((ab) ∩ (bb ))
3 an1 106 . . . 4 ((ab) ∩ 1) = (ab)
4 comor2 462 . . . . . 6 (ab) C b
54comcom2 183 . . . . . 6 (ab) C b
64, 5fh1 469 . . . . 5 ((ab) ∩ (bb )) = (((ab) ∩ b) ∪ ((ab) ∩ b ))
7 ax-a2 31 . . . . 5 (((ab) ∩ b) ∪ ((ab) ∩ b )) = (((ab) ∩ b ) ∪ ((ab) ∩ b))
8 mhlem1.1 . . . . . . . . . 10 a C b
98comcom2 183 . . . . . . . . 9 a C b
109comcom 453 . . . . . . . 8 b C a
11 comid 187 . . . . . . . . 9 b C b
1211comcom3 454 . . . . . . . 8 b C b
1310, 12fh1r 473 . . . . . . 7 ((ab) ∩ b ) = ((ab ) ∪ (bb ))
14 dff 101 . . . . . . . . 9 0 = (bb )
1514lor 70 . . . . . . . 8 ((ab ) ∪ 0) = ((ab ) ∪ (bb ))
1615ax-r1 35 . . . . . . 7 ((ab ) ∪ (bb )) = ((ab ) ∪ 0)
17 or0 102 . . . . . . 7 ((ab ) ∪ 0) = (ab )
1813, 16, 173tr 65 . . . . . 6 ((ab) ∩ b ) = (ab )
19 ancom 74 . . . . . . 7 ((ab) ∩ b) = (b ∩ (ab))
20 ax-a2 31 . . . . . . . 8 (ab) = (ba)
2120lan 77 . . . . . . 7 (b ∩ (ab)) = (b ∩ (ba))
22 anabs 121 . . . . . . 7 (b ∩ (ba)) = b
2319, 21, 223tr 65 . . . . . 6 ((ab) ∩ b) = b
2418, 232or 72 . . . . 5 (((ab) ∩ b ) ∪ ((ab) ∩ b)) = ((ab ) ∪ b)
256, 7, 243tr 65 . . . 4 ((ab) ∩ (bb )) = ((ab ) ∪ b)
262, 3, 253tr2 64 . . 3 (ab) = ((ab ) ∪ b)
2726ran 78 . 2 ((ab) ∩ (bc)) = (((ab ) ∪ b) ∩ (bc))
28 comorr 184 . . . . 5 b C (bc)
2928comcom6 459 . . . 4 b C (bc)
30 comanr2 465 . . . . 5 b C (ab )
3130comcom6 459 . . . 4 b C (ab )
3229, 31fh2rc 480 . . 3 (((ab ) ∪ b) ∩ (bc)) = (((ab ) ∩ (bc)) ∪ (b ∩ (bc)))
33 leao2 163 . . . . 5 (ab ) ≤ (bc)
3433df2le2 136 . . . 4 ((ab ) ∩ (bc)) = (ab )
3534ax-r5 38 . . 3 (((ab ) ∩ (bc)) ∪ (b ∩ (bc))) = ((ab ) ∪ (b ∩ (bc)))
3632, 35ax-r2 36 . 2 (((ab ) ∪ b) ∩ (bc)) = ((ab ) ∪ (b ∩ (bc)))
3711comcom2 183 . . . . 5 b C b
38 mhlem1.2 . . . . . 6 c C b
3938comcom 453 . . . . 5 b C c
4037, 39fh1 469 . . . 4 (b ∩ (bc)) = ((bb ) ∪ (bc))
4114ax-r5 38 . . . . 5 (0 ∪ (bc)) = ((bb ) ∪ (bc))
4241ax-r1 35 . . . 4 ((bb ) ∪ (bc)) = (0 ∪ (bc))
43 or0r 103 . . . 4 (0 ∪ (bc)) = (bc)
4440, 42, 433tr 65 . . 3 (b ∩ (bc)) = (bc)
4544lor 70 . 2 ((ab ) ∪ (b ∩ (bc))) = ((ab ) ∪ (bc))
4627, 36, 453tr 65 1 ((ab) ∩ (bc)) = ((ab ) ∪ (bc))
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  mhlem2  878
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