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Theorem wfh2 424
 Description: Weak structural analog of Foulis-Holland Theorem.
Hypotheses
Ref Expression
wfh.1 C (a, b) = 1
wfh.2 C (a, c) = 1
Assertion
Ref Expression
wfh2 ((b ∩ (ac)) ≡ ((ba) ∪ (bc))) = 1

Proof of Theorem wfh2
StepHypRef Expression
1 wledi 405 . . 3 (((ba) ∪ (bc)) ≤2 (b ∩ (ac))) = 1
2 oran 87 . . . . . . . . . . . 12 ((ba) ∪ (bc)) = ((ba) ∩ (bc) )
32bi1 118 . . . . . . . . . . 11 (((ba) ∪ (bc)) ≡ ((ba) ∩ (bc) ) ) = 1
4 df-a 40 . . . . . . . . . . . . . . 15 (ba) = (ba )
54bi1 118 . . . . . . . . . . . . . 14 ((ba) ≡ (ba ) ) = 1
65wcon2 208 . . . . . . . . . . . . 13 ((ba) ≡ (ba )) = 1
76wran 369 . . . . . . . . . . . 12 (((ba) ∩ (bc) ) ≡ ((ba ) ∩ (bc) )) = 1
87wr4 199 . . . . . . . . . . 11 (((ba) ∩ (bc) ) ≡ ((ba ) ∩ (bc) ) ) = 1
93, 8wr2 371 . . . . . . . . . 10 (((ba) ∪ (bc)) ≡ ((ba ) ∩ (bc) ) ) = 1
109wcon2 208 . . . . . . . . 9 (((ba) ∪ (bc)) ≡ ((ba ) ∩ (bc) )) = 1
1110wlan 370 . . . . . . . 8 (((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) ≡ ((b ∩ (ac)) ∩ ((ba ) ∩ (bc) ))) = 1
12 an4 86 . . . . . . . . . 10 ((b ∩ (ac)) ∩ ((ba ) ∩ (bc) )) = ((b ∩ (ba )) ∩ ((ac) ∩ (bc) ))
1312bi1 118 . . . . . . . . 9 (((b ∩ (ac)) ∩ ((ba ) ∩ (bc) )) ≡ ((b ∩ (ba )) ∩ ((ac) ∩ (bc) ))) = 1
14 wfh.1 . . . . . . . . . . . . . 14 C (a, b) = 1
1514wcomcom 414 . . . . . . . . . . . . 13 C (b, a) = 1
1615wcomcom2 415 . . . . . . . . . . . 12 C (b, a ) = 1
1716wcom3ii 419 . . . . . . . . . . 11 ((b ∩ (ba )) ≡ (ba )) = 1
18 ancom 74 . . . . . . . . . . . 12 (ba ) = (ab)
1918bi1 118 . . . . . . . . . . 11 ((ba ) ≡ (ab)) = 1
2017, 19wr2 371 . . . . . . . . . 10 ((b ∩ (ba )) ≡ (ab)) = 1
2120wran 369 . . . . . . . . 9 (((b ∩ (ba )) ∩ ((ac) ∩ (bc) )) ≡ ((ab) ∩ ((ac) ∩ (bc) ))) = 1
2213, 21wr2 371 . . . . . . . 8 (((b ∩ (ac)) ∩ ((ba ) ∩ (bc) )) ≡ ((ab) ∩ ((ac) ∩ (bc) ))) = 1
2311, 22wr2 371 . . . . . . 7 (((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) ≡ ((ab) ∩ ((ac) ∩ (bc) ))) = 1
24 an4 86 . . . . . . . 8 ((ab) ∩ ((ac) ∩ (bc) )) = ((a ∩ (ac)) ∩ (b ∩ (bc) ))
2524bi1 118 . . . . . . 7 (((ab) ∩ ((ac) ∩ (bc) )) ≡ ((a ∩ (ac)) ∩ (b ∩ (bc) ))) = 1
2623, 25wr2 371 . . . . . 6 (((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) ≡ ((a ∩ (ac)) ∩ (b ∩ (bc) ))) = 1
27 ax-a1 30 . . . . . . . . . . 11 a = a
2827bi1 118 . . . . . . . . . 10 (aa ) = 1
2928wr5-2v 366 . . . . . . . . 9 ((ac) ≡ (a c)) = 1
3029wlan 370 . . . . . . . 8 ((a ∩ (ac)) ≡ (a ∩ (a c))) = 1
31 wfh.2 . . . . . . . . . 10 C (a, c) = 1
3231wcomcom3 416 . . . . . . . . 9 C (a , c) = 1
3332wcom3ii 419 . . . . . . . 8 ((a ∩ (a c)) ≡ (ac)) = 1
3430, 33wr2 371 . . . . . . 7 ((a ∩ (ac)) ≡ (ac)) = 1
3534wran 369 . . . . . 6 (((a ∩ (ac)) ∩ (b ∩ (bc) )) ≡ ((ac) ∩ (b ∩ (bc) ))) = 1
3626, 35wr2 371 . . . . 5 (((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) ≡ ((ac) ∩ (b ∩ (bc) ))) = 1
37 anass 76 . . . . . 6 ((ac) ∩ (b ∩ (bc) )) = (a ∩ (c ∩ (b ∩ (bc) )))
3837bi1 118 . . . . 5 (((ac) ∩ (b ∩ (bc) )) ≡ (a ∩ (c ∩ (b ∩ (bc) )))) = 1
3936, 38wr2 371 . . . 4 (((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) ≡ (a ∩ (c ∩ (b ∩ (bc) )))) = 1
40 anass 76 . . . . . . . . 9 ((bc) ∩ (bc) ) = (b ∩ (c ∩ (bc) ))
4140bi1 118 . . . . . . . 8 (((bc) ∩ (bc) ) ≡ (b ∩ (c ∩ (bc) ))) = 1
4241wr1 197 . . . . . . 7 ((b ∩ (c ∩ (bc) )) ≡ ((bc) ∩ (bc) )) = 1
43 an12 81 . . . . . . . 8 (c ∩ (b ∩ (bc) )) = (b ∩ (c ∩ (bc) ))
4443bi1 118 . . . . . . 7 ((c ∩ (b ∩ (bc) )) ≡ (b ∩ (c ∩ (bc) ))) = 1
45 dff 101 . . . . . . . 8 0 = ((bc) ∩ (bc) )
4645bi1 118 . . . . . . 7 (0 ≡ ((bc) ∩ (bc) )) = 1
4742, 44, 46w3tr1 374 . . . . . 6 ((c ∩ (b ∩ (bc) )) ≡ 0) = 1
4847wlan 370 . . . . 5 ((a ∩ (c ∩ (b ∩ (bc) ))) ≡ (a ∩ 0)) = 1
49 an0 108 . . . . . 6 (a ∩ 0) = 0
5049bi1 118 . . . . 5 ((a ∩ 0) ≡ 0) = 1
5148, 50wr2 371 . . . 4 ((a ∩ (c ∩ (b ∩ (bc) ))) ≡ 0) = 1
5239, 51wr2 371 . . 3 (((b ∩ (ac)) ∩ ((ba) ∪ (bc)) ) ≡ 0) = 1
531, 52wom5 381 . 2 (((ba) ∪ (bc)) ≡ (b ∩ (ac))) = 1
5453wr1 197 1 ((b ∩ (ac)) ≡ ((ba) ∪ (bc))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   C wcmtr 29 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-wom 361 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by:  wfh4  426
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