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Theorem test 802
 Description: Part of an attempt to crack a potential Kalmbach axiom.
Assertion
Ref Expression
test (((c ∪ (ab )) ∩ (c ∩ (c ∪ (ab)))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))

Proof of Theorem test
StepHypRef Expression
1 oran3 93 . . . . 5 (ab ) = (ab)
21lor 70 . . . 4 (c ∪ (ab )) = (c ∪ (ab) )
32ran 78 . . 3 ((c ∪ (ab )) ∩ (c ∩ (c ∪ (ab)))) = ((c ∪ (ab) ) ∩ (c ∩ (c ∪ (ab))))
43ax-r5 38 . 2 (((c ∪ (ab )) ∩ (c ∩ (c ∪ (ab)))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = (((c ∪ (ab) ) ∩ (c ∩ (c ∪ (ab)))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab)))))
5 comor1 461 . . . . . . 7 (c ∪ (ab) ) C c
65comcom2 183 . . . . . 6 (c ∪ (ab) ) C c
7 comor2 462 . . . . . . 7 (c ∪ (ab) ) C (ab)
87comcom7 460 . . . . . 6 (c ∪ (ab) ) C (ab)
96, 8com2an 484 . . . . 5 (c ∪ (ab) ) C (c ∩ (ab))
106, 8com2or 483 . . . . . 6 (c ∪ (ab) ) C (c ∪ (ab))
115, 10com2an 484 . . . . 5 (c ∪ (ab) ) C (c ∩ (c ∪ (ab)))
129, 11com2or 483 . . . 4 (c ∪ (ab) ) C ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))
135, 8com2or 483 . . . . 5 (c ∪ (ab) ) C (c ∪ (ab))
146, 13com2an 484 . . . 4 (c ∪ (ab) ) C (c ∩ (c ∪ (ab)))
1512, 14fh4r 476 . . 3 (((c ∪ (ab) ) ∩ (c ∩ (c ∪ (ab)))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = (((c ∪ (ab) ) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) ∩ ((c ∩ (c ∪ (ab))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))))
16 ax-a3 32 . . . . . . 7 (((c ∪ (ab) ) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab)))) = ((c ∪ (ab) ) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab)))))
1716ax-r1 35 . . . . . 6 ((c ∪ (ab) ) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = (((c ∪ (ab) ) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab))))
18 ax-a2 31 . . . . . . 7 (((c ∪ (ab) ) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab)))) = ((c ∩ (c ∪ (ab))) ∪ ((c ∪ (ab) ) ∪ (c ∩ (ab))))
19 anor2 89 . . . . . . . . . . 11 (c ∩ (ab)) = (c ∪ (ab) )
2019lor 70 . . . . . . . . . 10 ((c ∪ (ab) ) ∪ (c ∩ (ab))) = ((c ∪ (ab) ) ∪ (c ∪ (ab) ) )
21 df-t 41 . . . . . . . . . . 11 1 = ((c ∪ (ab) ) ∪ (c ∪ (ab) ) )
2221ax-r1 35 . . . . . . . . . 10 ((c ∪ (ab) ) ∪ (c ∪ (ab) ) ) = 1
2320, 22ax-r2 36 . . . . . . . . 9 ((c ∪ (ab) ) ∪ (c ∩ (ab))) = 1
2423lor 70 . . . . . . . 8 ((c ∩ (c ∪ (ab))) ∪ ((c ∪ (ab) ) ∪ (c ∩ (ab)))) = ((c ∩ (c ∪ (ab))) ∪ 1)
25 or1 104 . . . . . . . 8 ((c ∩ (c ∪ (ab))) ∪ 1) = 1
2624, 25ax-r2 36 . . . . . . 7 ((c ∩ (c ∪ (ab))) ∪ ((c ∪ (ab) ) ∪ (c ∩ (ab)))) = 1
2718, 26ax-r2 36 . . . . . 6 (((c ∪ (ab) ) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab)))) = 1
2817, 27ax-r2 36 . . . . 5 ((c ∪ (ab) ) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = 1
29 ax-a3 32 . . . . . . 7 (((c ∩ (c ∪ (ab))) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab)))) = ((c ∩ (c ∪ (ab))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab)))))
3029ax-r1 35 . . . . . 6 ((c ∩ (c ∪ (ab))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = (((c ∩ (c ∪ (ab))) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab))))
31 ax-a2 31 . . . . . . . . 9 ((c ∩ (c ∪ (ab))) ∪ (c ∩ (ab))) = ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))
32 leor 159 . . . . . . . . . . 11 (ab) ≤ (c ∪ (ab))
3332lelan 167 . . . . . . . . . 10 (c ∩ (ab)) ≤ (c ∩ (c ∪ (ab)))
3433df-le2 131 . . . . . . . . 9 ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab)))) = (c ∩ (c ∪ (ab)))
3531, 34ax-r2 36 . . . . . . . 8 ((c ∩ (c ∪ (ab))) ∪ (c ∩ (ab))) = (c ∩ (c ∪ (ab)))
3635ax-r5 38 . . . . . . 7 (((c ∩ (c ∪ (ab))) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab)))) = ((c ∩ (c ∪ (ab))) ∪ (c ∩ (c ∪ (ab))))
37 coman1 185 . . . . . . . . . 10 (c ∩ (c ∪ (ab))) C c
3837comcom7 460 . . . . . . . . 9 (c ∩ (c ∪ (ab))) C c
39 comor1 461 . . . . . . . . . . 11 (c ∪ (ab)) C c
4039comcom7 460 . . . . . . . . . . . 12 (c ∪ (ab)) C c
41 comor2 462 . . . . . . . . . . . 12 (c ∪ (ab)) C (ab)
4240, 41com2or 483 . . . . . . . . . . 11 (c ∪ (ab)) C (c ∪ (ab))
4339, 42com2an 484 . . . . . . . . . 10 (c ∪ (ab)) C (c ∩ (c ∪ (ab)))
4443comcom 453 . . . . . . . . 9 (c ∩ (c ∪ (ab))) C (c ∪ (ab))
4538, 44fh3 471 . . . . . . . 8 ((c ∩ (c ∪ (ab))) ∪ (c ∩ (c ∪ (ab)))) = (((c ∩ (c ∪ (ab))) ∪ c) ∩ ((c ∩ (c ∪ (ab))) ∪ (c ∪ (ab))))
46 ax-a2 31 . . . . . . . . . 10 ((c ∩ (c ∪ (ab))) ∪ c) = (c ∪ (c ∩ (c ∪ (ab))))
47 oml 445 . . . . . . . . . 10 (c ∪ (c ∩ (c ∪ (ab)))) = (c ∪ (ab))
4846, 47ax-r2 36 . . . . . . . . 9 ((c ∩ (c ∪ (ab))) ∪ c) = (c ∪ (ab))
49 or12 80 . . . . . . . . . 10 ((c ∩ (c ∪ (ab))) ∪ (c ∪ (ab))) = (c ∪ ((c ∩ (c ∪ (ab))) ∪ (ab)))
50 ax-a3 32 . . . . . . . . . . . 12 ((c ∪ (c ∩ (c ∪ (ab)))) ∪ (ab)) = (c ∪ ((c ∩ (c ∪ (ab))) ∪ (ab)))
5150ax-r1 35 . . . . . . . . . . 11 (c ∪ ((c ∩ (c ∪ (ab))) ∪ (ab))) = ((c ∪ (c ∩ (c ∪ (ab)))) ∪ (ab))
52 orabs 120 . . . . . . . . . . . 12 (c ∪ (c ∩ (c ∪ (ab)))) = c
5352ax-r5 38 . . . . . . . . . . 11 ((c ∪ (c ∩ (c ∪ (ab)))) ∪ (ab)) = (c ∪ (ab))
5451, 53ax-r2 36 . . . . . . . . . 10 (c ∪ ((c ∩ (c ∪ (ab))) ∪ (ab))) = (c ∪ (ab))
5549, 54ax-r2 36 . . . . . . . . 9 ((c ∩ (c ∪ (ab))) ∪ (c ∪ (ab))) = (c ∪ (ab))
5648, 552an 79 . . . . . . . 8 (((c ∩ (c ∪ (ab))) ∪ c) ∩ ((c ∩ (c ∪ (ab))) ∪ (c ∪ (ab)))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
5745, 56ax-r2 36 . . . . . . 7 ((c ∩ (c ∪ (ab))) ∪ (c ∩ (c ∪ (ab)))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
5836, 57ax-r2 36 . . . . . 6 (((c ∩ (c ∪ (ab))) ∪ (c ∩ (ab))) ∪ (c ∩ (c ∪ (ab)))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
5930, 58ax-r2 36 . . . . 5 ((c ∩ (c ∪ (ab))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
6028, 592an 79 . . . 4 (((c ∪ (ab) ) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) ∩ ((c ∩ (c ∪ (ab))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab)))))) = (1 ∩ ((c ∪ (ab)) ∩ (c ∪ (ab))))
61 ancom 74 . . . . 5 (1 ∩ ((c ∪ (ab)) ∩ (c ∪ (ab)))) = (((c ∪ (ab)) ∩ (c ∪ (ab))) ∩ 1)
62 an1 106 . . . . 5 (((c ∪ (ab)) ∩ (c ∪ (ab))) ∩ 1) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
6361, 62ax-r2 36 . . . 4 (1 ∩ ((c ∪ (ab)) ∩ (c ∪ (ab)))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
6460, 63ax-r2 36 . . 3 (((c ∪ (ab) ) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) ∩ ((c ∩ (c ∪ (ab))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab)))))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
6515, 64ax-r2 36 . 2 (((c ∪ (ab) ) ∩ (c ∩ (c ∪ (ab)))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
664, 65ax-r2 36 1 (((c ∪ (ab )) ∩ (c ∩ (c ∪ (ab)))) ∪ ((c ∩ (ab)) ∪ (c ∩ (c ∪ (ab))))) = ((c ∪ (ab)) ∩ (c ∪ (ab)))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8 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: (None)
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