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Theorem oa3-2lemb 979
 Description: Lemma for 3-OA(2). Equivalence with substitution into 4-OA.
Assertion
Ref Expression
oa3-2lemb ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))

Proof of Theorem oa3-2lemb
StepHypRef Expression
1 ax-a3 32 . . . . 5 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c))))) = ((ab) ∪ (((a1 c) ∩ (b1 c)) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c))))))
2 i1id 275 . . . . . . . . . . . . 13 (c1 c) = 1
32lan 77 . . . . . . . . . . . 12 ((a1 c) ∩ (c1 c)) = ((a1 c) ∩ 1)
4 an1 106 . . . . . . . . . . . 12 ((a1 c) ∩ 1) = (a1 c)
53, 4ax-r2 36 . . . . . . . . . . 11 ((a1 c) ∩ (c1 c)) = (a1 c)
65lor 70 . . . . . . . . . 10 ((ac) ∪ ((a1 c) ∩ (c1 c))) = ((ac) ∪ (a1 c))
7 or12 80 . . . . . . . . . . . 12 ((ac) ∪ (a ∪ (ac))) = (a ∪ ((ac) ∪ (ac)))
8 oridm 110 . . . . . . . . . . . . 13 ((ac) ∪ (ac)) = (ac)
98lor 70 . . . . . . . . . . . 12 (a ∪ ((ac) ∪ (ac))) = (a ∪ (ac))
107, 9ax-r2 36 . . . . . . . . . . 11 ((ac) ∪ (a ∪ (ac))) = (a ∪ (ac))
11 df-i1 44 . . . . . . . . . . . 12 (a1 c) = (a ∪ (ac))
1211lor 70 . . . . . . . . . . 11 ((ac) ∪ (a1 c)) = ((ac) ∪ (a ∪ (ac)))
1310, 12, 113tr1 63 . . . . . . . . . 10 ((ac) ∪ (a1 c)) = (a1 c)
146, 13ax-r2 36 . . . . . . . . 9 ((ac) ∪ ((a1 c) ∩ (c1 c))) = (a1 c)
152lan 77 . . . . . . . . . . . 12 ((b1 c) ∩ (c1 c)) = ((b1 c) ∩ 1)
16 an1 106 . . . . . . . . . . . 12 ((b1 c) ∩ 1) = (b1 c)
1715, 16ax-r2 36 . . . . . . . . . . 11 ((b1 c) ∩ (c1 c)) = (b1 c)
1817lor 70 . . . . . . . . . 10 ((bc) ∪ ((b1 c) ∩ (c1 c))) = ((bc) ∪ (b1 c))
19 or12 80 . . . . . . . . . . . 12 ((bc) ∪ (b ∪ (bc))) = (b ∪ ((bc) ∪ (bc)))
20 oridm 110 . . . . . . . . . . . . 13 ((bc) ∪ (bc)) = (bc)
2120lor 70 . . . . . . . . . . . 12 (b ∪ ((bc) ∪ (bc))) = (b ∪ (bc))
2219, 21ax-r2 36 . . . . . . . . . . 11 ((bc) ∪ (b ∪ (bc))) = (b ∪ (bc))
23 df-i1 44 . . . . . . . . . . . 12 (b1 c) = (b ∪ (bc))
2423lor 70 . . . . . . . . . . 11 ((bc) ∪ (b1 c)) = ((bc) ∪ (b ∪ (bc)))
2522, 24, 233tr1 63 . . . . . . . . . 10 ((bc) ∪ (b1 c)) = (b1 c)
2618, 25ax-r2 36 . . . . . . . . 9 ((bc) ∪ ((b1 c) ∩ (c1 c))) = (b1 c)
2714, 262an 79 . . . . . . . 8 (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))) = ((a1 c) ∩ (b1 c))
2827lor 70 . . . . . . 7 (((a1 c) ∩ (b1 c)) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c))))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
29 oridm 110 . . . . . . 7 (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))) = ((a1 c) ∩ (b1 c))
3028, 29ax-r2 36 . . . . . 6 (((a1 c) ∩ (b1 c)) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c))))) = ((a1 c) ∩ (b1 c))
3130lor 70 . . . . 5 ((ab) ∪ (((a1 c) ∩ (b1 c)) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
321, 31ax-r2 36 . . . 4 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c))))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
3332lan 77 . . 3 (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))))) = (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))
3433lor 70 . 2 (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c))))))) = (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))
3534lan 77 1 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((ac) ∪ ((a1 c) ∩ (c1 c))) ∩ ((bc) ∪ ((b1 c) ∩ (c1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44 This theorem is referenced by:  oa3-2to4  988
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