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Theorem lem4.6.7 1101
 Description: Equation 4.15 of [MegPav2000] p. 23. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
lem4.6.7.1 ab
Assertion
Ref Expression
lem4.6.7 b ≤ (a1 b)

Proof of Theorem lem4.6.7
StepHypRef Expression
1 leid 148 . . . . . . 7 aa
21sklem 230 . . . . . 6 (aa) = 1
32ax-r1 35 . . . . 5 1 = (aa)
4 lem4.6.7.1 . . . . . . 7 ab
54df-le2 131 . . . . . 6 (ab) = b
65ax-r1 35 . . . . 5 b = (ab)
73, 62an 79 . . . 4 (1 ∩ b) = ((aa) ∩ (ab))
8 ax-a3 32 . . . . . 6 ((ba ) ∪ (ab)) = (b ∪ (a ∪ (ab)))
98ax-r1 35 . . . . 5 (b ∪ (a ∪ (ab))) = ((ba ) ∪ (ab))
10 le1 146 . . . . . . . . 9 b ≤ 1
11 leid 148 . . . . . . . . 9 bb
1210, 11ler2an 173 . . . . . . . 8 b ≤ (1 ∩ b)
13 le1 146 . . . . . . . . 9 a ≤ 1
1413, 4ler2an 173 . . . . . . . 8 a ≤ (1 ∩ b)
1512, 14lel2or 170 . . . . . . 7 (ba ) ≤ (1 ∩ b)
16 le1 146 . . . . . . . 8 a ≤ 1
1716leran 153 . . . . . . 7 (ab) ≤ (1 ∩ b)
1815, 17lel2or 170 . . . . . 6 ((ba ) ∪ (ab)) ≤ (1 ∩ b)
19 leao2 163 . . . . . . 7 (1 ∩ b) ≤ (ba )
2019ler 149 . . . . . 6 (1 ∩ b) ≤ ((ba ) ∪ (ab))
2118, 20lebi 145 . . . . 5 ((ba ) ∪ (ab)) = (1 ∩ b)
229, 21ax-r2 36 . . . 4 (b ∪ (a ∪ (ab))) = (1 ∩ b)
23 comid 187 . . . . . 6 a C a
2423comcom3 454 . . . . 5 a C a
254lecom 180 . . . . 5 a C b
2624, 25fh3 471 . . . 4 (a ∪ (ab)) = ((aa) ∩ (ab))
277, 22, 263tr1 63 . . 3 (b ∪ (a ∪ (ab))) = (a ∪ (ab))
2827df-le1 130 . 2 b ≤ (a ∪ (ab))
29 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
3029ax-r1 35 . 2 (a ∪ (ab)) = (a1 b)
3128, 30lbtr 139 1 b ≤ (a1 b)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ 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: (None)
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