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Theorem wfh1 423
Description: Weak structural analog of Foulis-Holland Theorem.
Hypotheses
Ref Expression
wfh.1 C (a, b) = 1
wfh.2 C (a, c) = 1
Assertion
Ref Expression
wfh1 ((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1
Proof of Theorem wfh1
StepHypRef Expression
1 wledi 405 . . 3 (((a ^ b) v (a ^ c)) =<2 (a ^ (b v c))) = 1
2 ancom 74 . . . . . . 7 (a ^ (b v c)) = ((b v c) ^ a)
32bi1 118 . . . . . 6 ((a ^ (b v c)) == ((b v c) ^ a)) = 1
4 df-a 40 . . . . . . . . . 10 (a ^ b) = (a' v b')'
54bi1 118 . . . . . . . . 9 ((a ^ b) == (a' v b')') = 1
6 df-a 40 . . . . . . . . . 10 (a ^ c) = (a' v c')'
76bi1 118 . . . . . . . . 9 ((a ^ c) == (a' v c')') = 1
85, 7w2or 372 . . . . . . . 8 (((a ^ b) v (a ^ c)) == ((a' v b')' v (a' v c')')) = 1
9 df-a 40 . . . . . . . . . . 11 ((a' v b') ^ (a' v c')) = ((a' v b')' v (a' v c')')'
109bi1 118 . . . . . . . . . 10 (((a' v b') ^ (a' v c')) == ((a' v b')' v (a' v c')')') = 1
1110wr1 197 . . . . . . . . 9 (((a' v b')' v (a' v c')')' == ((a' v b') ^ (a' v c'))) = 1
1211wcon3 209 . . . . . . . 8 (((a' v b')' v (a' v c')') == ((a' v b') ^ (a' v c'))') = 1
138, 12wr2 371 . . . . . . 7 (((a ^ b) v (a ^ c)) == ((a' v b') ^ (a' v c'))') = 1
1413wcon2 208 . . . . . 6 (((a ^ b) v (a ^ c))' == ((a' v b') ^ (a' v c'))) = 1
153, 14w2an 373 . . . . 5 (((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))') == (((b v c) ^ a) ^ ((a' v b') ^ (a' v c')))) = 1
16 anass 76 . . . . . . . 8 (((b v c) ^ a) ^ ((a' v b') ^ (a' v c'))) = ((b v c) ^ (a ^ ((a' v b') ^ (a' v c'))))
1716bi1 118 . . . . . . 7 ((((b v c) ^ a) ^ ((a' v b') ^ (a' v c'))) == ((b v c) ^ (a ^ ((a' v b') ^ (a' v c'))))) = 1
18 wfh.1 . . . . . . . . . . . 12 C (a, b) = 1
1918wcomcom2 415 . . . . . . . . . . 11 C (a, b') = 1
2019wcom3ii 419 . . . . . . . . . 10 ((a ^ (a' v b')) == (a ^ b')) = 1
21 wfh.2 . . . . . . . . . . . 12 C (a, c) = 1
2221wcomcom2 415 . . . . . . . . . . 11 C (a, c') = 1
2322wcom3ii 419 . . . . . . . . . 10 ((a ^ (a' v c')) == (a ^ c')) = 1
2420, 23w2an 373 . . . . . . . . 9 (((a ^ (a' v b')) ^ (a ^ (a' v c'))) == ((a ^ b') ^ (a ^ c'))) = 1
25 anandi 114 . . . . . . . . . 10 (a ^ ((a' v b') ^ (a' v c'))) = ((a ^ (a' v b')) ^ (a ^ (a' v c')))
2625bi1 118 . . . . . . . . 9 ((a ^ ((a' v b') ^ (a' v c'))) == ((a ^ (a' v b')) ^ (a ^ (a' v c')))) = 1
27 anandi 114 . . . . . . . . . 10 (a ^ (b' ^ c')) = ((a ^ b') ^ (a ^ c'))
2827bi1 118 . . . . . . . . 9 ((a ^ (b' ^ c')) == ((a ^ b') ^ (a ^ c'))) = 1
2924, 26, 28w3tr1 374 . . . . . . . 8 ((a ^ ((a' v b') ^ (a' v c'))) == (a ^ (b' ^ c'))) = 1
3029wlan 370 . . . . . . 7 (((b v c) ^ (a ^ ((a' v b') ^ (a' v c')))) == ((b v c) ^ (a ^ (b' ^ c')))) = 1
3117, 30wr2 371 . . . . . 6 ((((b v c) ^ a) ^ ((a' v b') ^ (a' v c'))) == ((b v c) ^ (a ^ (b' ^ c')))) = 1
32 an12 81 . . . . . . 7 ((b v c) ^ (a ^ (b' ^ c'))) = (a ^ ((b v c) ^ (b' ^ c')))
3332bi1 118 . . . . . 6 (((b v c) ^ (a ^ (b' ^ c'))) == (a ^ ((b v c) ^ (b' ^ c')))) = 1
3431, 33wr2 371 . . . . 5 ((((b v c) ^ a) ^ ((a' v b') ^ (a' v c'))) == (a ^ ((b v c) ^ (b' ^ c')))) = 1
3515, 34wr2 371 . . . 4 (((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))') == (a ^ ((b v c) ^ (b' ^ c')))) = 1
36 oran 87 . . . . . . . . . . 11 (b v c) = (b' ^ c')'
3736bi1 118 . . . . . . . . . 10 ((b v c) == (b' ^ c')') = 1
3837wr1 197 . . . . . . . . 9 ((b' ^ c')' == (b v c)) = 1
3938wcon3 209 . . . . . . . 8 ((b' ^ c') == (b v c)') = 1
4039wlan 370 . . . . . . 7 (((b v c) ^ (b' ^ c')) == ((b v c) ^ (b v c)')) = 1
41 dff 101 . . . . . . . . 9 0 = ((b v c) ^ (b v c)')
4241bi1 118 . . . . . . . 8 (0 == ((b v c) ^ (b v c)')) = 1
4342wr1 197 . . . . . . 7 (((b v c) ^ (b v c)') == 0) = 1
4440, 43wr2 371 . . . . . 6 (((b v c) ^ (b' ^ c')) == 0) = 1
4544wlan 370 . . . . 5 ((a ^ ((b v c) ^ (b' ^ c'))) == (a ^ 0)) = 1
46 an0 108 . . . . . 6 (a ^ 0) = 0
4746bi1 118 . . . . 5 ((a ^ 0) == 0) = 1
4845, 47wr2 371 . . . 4 ((a ^ ((b v c) ^ (b' ^ c'))) == 0) = 1
4935, 48wr2 371 . . 3 (((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))') == 0) = 1
501, 49wom5 381 . 2 (((a ^ b) v (a ^ c)) == (a ^ (b v c))) = 1
5150wr1 197 1 ((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8  0wf 9   C wcmtr 29
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-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by:  wfh3  425  wcom2or  427  wnbdi  429  wlem14  430  ska2  432  wddi1  1105
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