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Theorem k1-7 354
Description: Statement (7) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.
Hypothesis
Ref Expression
k1-7.1 x = ((x ^ c) v (x ^ c'))
Assertion
Ref Expression
k1-7 (x' ^ c') = ((x' v c) ^ c')
Proof of Theorem k1-7
StepHypRef Expression
1 anor3 90 . . . . 5 ((x ^ c')' ^ (x ^ c'')') = ((x ^ c') v (x ^ c''))'
21cm 61 . . . 4 ((x ^ c') v (x ^ c''))' = ((x ^ c')' ^ (x ^ c'')')
3 k1-7.1 . . . . . 6 x = ((x ^ c) v (x ^ c'))
4 ax-a1 30 . . . . . . . 8 c = c''
54lan 77 . . . . . . 7 (x ^ c) = (x ^ c'')
65ror 71 . . . . . 6 ((x ^ c) v (x ^ c')) = ((x ^ c'') v (x ^ c'))
7 orcom 73 . . . . . 6 ((x ^ c'') v (x ^ c')) = ((x ^ c') v (x ^ c''))
83, 6, 73tr 65 . . . . 5 x = ((x ^ c') v (x ^ c''))
98con4 69 . . . 4 x' = ((x ^ c') v (x ^ c''))'
10 oran3 93 . . . . 5 (x' v c'') = (x ^ c')'
11 oran2 92 . . . . 5 (x' v c') = (x ^ c'')'
1210, 112an 79 . . . 4 ((x' v c'') ^ (x' v c')) = ((x ^ c')' ^ (x ^ c'')')
132, 9, 123tr1 63 . . 3 x' = ((x' v c'') ^ (x' v c'))
1413ran 78 . 2 (x' ^ c') = (((x' v c'') ^ (x' v c')) ^ c')
154lor 70 . . . . . 6 (x' v c) = (x' v c'')
1615ran 78 . . . . 5 ((x' v c) ^ (x' v c')) = ((x' v c'') ^ (x' v c'))
1716ran 78 . . . 4 (((x' v c) ^ (x' v c')) ^ c') = (((x' v c'') ^ (x' v c')) ^ c')
1817cm 61 . . 3 (((x' v c'') ^ (x' v c')) ^ c') = (((x' v c) ^ (x' v c')) ^ c')
19 anass 76 . . 3 (((x' v c) ^ (x' v c')) ^ c') = ((x' v c) ^ ((x' v c') ^ c'))
2018, 19tr 62 . 2 (((x' v c'') ^ (x' v c')) ^ c') = ((x' v c) ^ ((x' v c') ^ c'))
21 ancom 74 . . . 4 ((x' v c') ^ c') = (c' ^ (x' v c'))
22 ax-a2 31 . . . . 5 (x' v c') = (c' v x')
2322lan 77 . . . 4 (c' ^ (x' v c')) = (c' ^ (c' v x'))
24 anabs 121 . . . 4 (c' ^ (c' v x')) = c'
2521, 23, 243tr 65 . . 3 ((x' v c') ^ c') = c'
2625lan 77 . 2 ((x' v c) ^ ((x' v c') ^ c')) = ((x' v c) ^ c')
2714, 20, 263tr 65 1 (x' ^ c') = ((x' v c) ^ c')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7
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
This theorem is referenced by: (None)
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