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Theorem oa3-6lem 980
 Description: Lemma for 3-OA(6). Equivalence with substitution into 4-OA. (Contributed by NM, 24-Dec-1998.)
Assertion
Ref Expression
oa3-6lem ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))))

Proof of Theorem oa3-6lem
StepHypRef Expression
1 an1 106 . . . . . . . . 9 (a ∩ 1) = a
2 1i1 274 . . . . . . . . . . 11 (1 →1 c) = c
32lan 77 . . . . . . . . . 10 ((a1 c) ∩ (1 →1 c)) = ((a1 c) ∩ c)
4 u1lemab 610 . . . . . . . . . 10 ((a1 c) ∩ c) = ((ac) ∪ (ac))
53, 4ax-r2 36 . . . . . . . . 9 ((a1 c) ∩ (1 →1 c)) = ((ac) ∪ (ac))
61, 52or 72 . . . . . . . 8 ((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) = (a ∪ ((ac) ∪ (ac)))
7 ax-a3 32 . . . . . . . . 9 ((a ∪ (ac)) ∪ (ac)) = (a ∪ ((ac) ∪ (ac)))
87ax-r1 35 . . . . . . . 8 (a ∪ ((ac) ∪ (ac))) = ((a ∪ (ac)) ∪ (ac))
9 orabs 120 . . . . . . . . 9 (a ∪ (ac)) = a
109ax-r5 38 . . . . . . . 8 ((a ∪ (ac)) ∪ (ac)) = (a ∪ (ac))
116, 8, 103tr 65 . . . . . . 7 ((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) = (a ∪ (ac))
12 an1 106 . . . . . . . . 9 (b ∩ 1) = b
132lan 77 . . . . . . . . . 10 ((b1 c) ∩ (1 →1 c)) = ((b1 c) ∩ c)
14 u1lemab 610 . . . . . . . . . 10 ((b1 c) ∩ c) = ((bc) ∪ (bc))
1513, 14ax-r2 36 . . . . . . . . 9 ((b1 c) ∩ (1 →1 c)) = ((bc) ∪ (bc))
1612, 152or 72 . . . . . . . 8 ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c))) = (b ∪ ((bc) ∪ (bc)))
17 ax-a3 32 . . . . . . . . 9 ((b ∪ (bc)) ∪ (bc)) = (b ∪ ((bc) ∪ (bc)))
1817ax-r1 35 . . . . . . . 8 (b ∪ ((bc) ∪ (bc))) = ((b ∪ (bc)) ∪ (bc))
19 orabs 120 . . . . . . . . 9 (b ∪ (bc)) = b
2019ax-r5 38 . . . . . . . 8 ((b ∪ (bc)) ∪ (bc)) = (b ∪ (bc))
2116, 18, 203tr 65 . . . . . . 7 ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c))) = (b ∪ (bc))
2211, 212an 79 . . . . . 6 (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c)))) = ((a ∪ (ac)) ∩ (b ∪ (bc)))
2322lor 70 . . . . 5 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c))))) = (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ ((a ∪ (ac)) ∩ (b ∪ (bc))))
24 or32 82 . . . . 5 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ ((a ∪ (ac)) ∩ (b ∪ (bc)))) = (((ab) ∪ ((a ∪ (ac)) ∩ (b ∪ (bc)))) ∪ ((a1 c) ∩ (b1 c)))
25 leo 158 . . . . . . . . 9 a ≤ (a ∪ (ac))
26 leo 158 . . . . . . . . 9 b ≤ (b ∪ (bc))
2725, 26le2an 169 . . . . . . . 8 (ab) ≤ ((a ∪ (ac)) ∩ (b ∪ (bc)))
2827df-le2 131 . . . . . . 7 ((ab) ∪ ((a ∪ (ac)) ∩ (b ∪ (bc)))) = ((a ∪ (ac)) ∩ (b ∪ (bc)))
29 ax-a1 30 . . . . . . . . . 10 a = a
3029ax-r5 38 . . . . . . . . 9 (a ∪ (ac)) = (a ∪ (ac))
31 df-i1 44 . . . . . . . . . 10 (a1 c) = (a ∪ (ac))
3231ax-r1 35 . . . . . . . . 9 (a ∪ (ac)) = (a1 c)
3330, 32ax-r2 36 . . . . . . . 8 (a ∪ (ac)) = (a1 c)
34 ax-a1 30 . . . . . . . . . 10 b = b
3534ax-r5 38 . . . . . . . . 9 (b ∪ (bc)) = (b ∪ (bc))
36 df-i1 44 . . . . . . . . . 10 (b1 c) = (b ∪ (bc))
3736ax-r1 35 . . . . . . . . 9 (b ∪ (bc)) = (b1 c)
3835, 37ax-r2 36 . . . . . . . 8 (b ∪ (bc)) = (b1 c)
3933, 382an 79 . . . . . . 7 ((a ∪ (ac)) ∩ (b ∪ (bc))) = ((a1 c) ∩ (b1 c))
4028, 39ax-r2 36 . . . . . 6 ((ab) ∪ ((a ∪ (ac)) ∩ (b ∪ (bc)))) = ((a1 c) ∩ (b1 c))
4140ax-r5 38 . . . . 5 (((ab) ∪ ((a ∪ (ac)) ∩ (b ∪ (bc)))) ∪ ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
4223, 24, 413tr 65 . . . 4 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c))))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
4342lan 77 . . 3 (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c)))))) = (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))
4443lor 70 . 2 (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c))))))) = (a ∪ (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))
4544lan 77 1 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((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-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:  oa3-6to3  987
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