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Theorem lem4.6.2e1 1080
 Description: Equation 4.10 of [MegPav2000] p. 23. This is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem4.6.2e1 ((a1 b) ∩ (a1 b)) = ((a1 b) ∩ b)

Proof of Theorem lem4.6.2e1
StepHypRef Expression
1 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
2 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
31, 22an 79 . 2 ((a1 b) ∩ (a1 b)) = ((a ∪ (ab)) ∩ (a ∪ (ab)))
4 ax-a1 30 . . . . . 6 a = a
54ax-r1 35 . . . . 5 a = a
65ax-r5 38 . . . 4 (a ∪ (ab)) = (a ∪ (ab))
76lan 77 . . 3 ((a ∪ (ab)) ∩ (a ∪ (ab))) = ((a ∪ (ab)) ∩ (a ∪ (ab)))
8 comorr 184 . . . . . 6 a C (a ∪ (ab))
98comcom6 459 . . . . 5 a C (a ∪ (ab))
109comcom 453 . . . 4 (a ∪ (ab)) C a
11 leao1 162 . . . . . 6 (ab) ≤ (a ∪ (ab))
1211lecom 180 . . . . 5 (ab) C (a ∪ (ab))
1312comcom 453 . . . 4 (a ∪ (ab)) C (ab)
1410, 13fh1 469 . . 3 ((a ∪ (ab)) ∩ (a ∪ (ab))) = (((a ∪ (ab)) ∩ a) ∪ ((a ∪ (ab)) ∩ (ab)))
15 ancom 74 . . . . 5 ((a ∪ (ab)) ∩ a) = (a ∩ (a ∪ (ab)))
1615ax-r5 38 . . . 4 (((a ∪ (ab)) ∩ a) ∪ ((a ∪ (ab)) ∩ (ab))) = ((a ∩ (a ∪ (ab))) ∪ ((a ∪ (ab)) ∩ (ab)))
17 omla 447 . . . . 5 (a ∩ (a ∪ (ab))) = (ab)
1817ax-r5 38 . . . 4 ((a ∩ (a ∪ (ab))) ∪ ((a ∪ (ab)) ∩ (ab))) = ((ab) ∪ ((a ∪ (ab)) ∩ (ab)))
19 ancom 74 . . . . . 6 ((a ∪ (ab)) ∩ (ab)) = ((ab) ∩ (a ∪ (ab)))
2019lor 70 . . . . 5 ((ab) ∪ ((a ∪ (ab)) ∩ (ab))) = ((ab) ∪ ((ab) ∩ (a ∪ (ab))))
21 coman1 185 . . . . . . 7 (ab) C a
2221comcom7 460 . . . . . . . 8 (ab) C a
23 coman2 186 . . . . . . . 8 (ab) C b
2422, 23com2an 484 . . . . . . 7 (ab) C (ab)
2521, 24fh1 469 . . . . . 6 ((ab) ∩ (a ∪ (ab))) = (((ab) ∩ a ) ∪ ((ab) ∩ (ab)))
2625lor 70 . . . . 5 ((ab) ∪ ((ab) ∩ (a ∪ (ab)))) = ((ab) ∪ (((ab) ∩ a ) ∪ ((ab) ∩ (ab))))
27 ancom 74 . . . . . . . . 9 (ab) = (ba )
2827ran 78 . . . . . . . 8 ((ab) ∩ a ) = ((ba ) ∩ a )
2928ax-r5 38 . . . . . . 7 (((ab) ∩ a ) ∪ ((ab) ∩ (ab))) = (((ba ) ∩ a ) ∪ ((ab) ∩ (ab)))
3029lor 70 . . . . . 6 ((ab) ∪ (((ab) ∩ a ) ∪ ((ab) ∩ (ab)))) = ((ab) ∪ (((ba ) ∩ a ) ∪ ((ab) ∩ (ab))))
31 anass 76 . . . . . . . 8 ((ba ) ∩ a ) = (b ∩ (aa ))
3231ax-r5 38 . . . . . . 7 (((ba ) ∩ a ) ∪ ((ab) ∩ (ab))) = ((b ∩ (aa )) ∪ ((ab) ∩ (ab)))
3332lor 70 . . . . . 6 ((ab) ∪ (((ba ) ∩ a ) ∪ ((ab) ∩ (ab)))) = ((ab) ∪ ((b ∩ (aa )) ∪ ((ab) ∩ (ab))))
34 anidm 111 . . . . . . . . . 10 (aa ) = a
3534lan 77 . . . . . . . . 9 (b ∩ (aa )) = (ba )
3635ax-r5 38 . . . . . . . 8 ((b ∩ (aa )) ∪ ((ab) ∩ (ab))) = ((ba ) ∪ ((ab) ∩ (ab)))
3736lor 70 . . . . . . 7 ((ab) ∪ ((b ∩ (aa )) ∪ ((ab) ∩ (ab)))) = ((ab) ∪ ((ba ) ∪ ((ab) ∩ (ab))))
38 ancom 74 . . . . . . . . 9 (ba ) = (ab)
3938ax-r5 38 . . . . . . . 8 ((ba ) ∪ ((ab) ∩ (ab))) = ((ab) ∪ ((ab) ∩ (ab)))
4039lor 70 . . . . . . 7 ((ab) ∪ ((ba ) ∪ ((ab) ∩ (ab)))) = ((ab) ∪ ((ab) ∪ ((ab) ∩ (ab))))
41 orabs 120 . . . . . . . . 9 ((ab) ∪ ((ab) ∩ (ab))) = (ab)
4241lor 70 . . . . . . . 8 ((ab) ∪ ((ab) ∪ ((ab) ∩ (ab)))) = ((ab) ∪ (ab))
43 coman1 185 . . . . . . . . . 10 (ab) C a
4443comcom2 183 . . . . . . . . 9 (ab) C a
45 coman2 186 . . . . . . . . 9 (ab) C b
4644, 45fh3 471 . . . . . . . 8 ((ab) ∪ (ab)) = (((ab) ∪ a ) ∩ ((ab) ∪ b))
47 ax-a2 31 . . . . . . . . 9 ((ab) ∪ a ) = (a ∪ (ab))
48 lear 161 . . . . . . . . . 10 (ab) ≤ b
4948df-le2 131 . . . . . . . . 9 ((ab) ∪ b) = b
5047, 492an 79 . . . . . . . 8 (((ab) ∪ a ) ∩ ((ab) ∪ b)) = ((a ∪ (ab)) ∩ b)
5142, 46, 503tr 65 . . . . . . 7 ((ab) ∪ ((ab) ∪ ((ab) ∩ (ab)))) = ((a ∪ (ab)) ∩ b)
5237, 40, 513tr 65 . . . . . 6 ((ab) ∪ ((b ∩ (aa )) ∪ ((ab) ∩ (ab)))) = ((a ∪ (ab)) ∩ b)
5330, 33, 523tr 65 . . . . 5 ((ab) ∪ (((ab) ∩ a ) ∪ ((ab) ∩ (ab)))) = ((a ∪ (ab)) ∩ b)
5420, 26, 533tr 65 . . . 4 ((ab) ∪ ((a ∪ (ab)) ∩ (ab))) = ((a ∪ (ab)) ∩ b)
5516, 18, 543tr 65 . . 3 (((a ∪ (ab)) ∩ a) ∪ ((a ∪ (ab)) ∩ (ab))) = ((a ∪ (ab)) ∩ b)
567, 14, 553tr 65 . 2 ((a ∪ (ab)) ∩ (a ∪ (ab))) = ((a ∪ (ab)) ∩ b)
571ax-r1 35 . . 3 (a ∪ (ab)) = (a1 b)
5857ran 78 . 2 ((a ∪ (ab)) ∩ b) = ((a1 b) ∩ b)
593, 56, 583tr 65 1 ((a1 b) ∩ (a1 b)) = ((a1 b) ∩ b)
 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: (None)
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