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Theorem gsth 489
Description: Gudder-Schelp's Theorem. Beran, p. 262, Th. 4.1.
Hypotheses
Ref Expression
gsth.1 a C b
gsth.2 b C c
gsth.3 a C (b ^ c)
Assertion
Ref Expression
gsth (a ^ b) C c
Proof of Theorem gsth
StepHypRef Expression
1 gsth.2 . . . . . 6 b C c
2 gsth.1 . . . . . . 7 a C b
32comcom 453 . . . . . 6 b C a
41, 3fh4rc 482 . . . . 5 ((a ^ b) v c) = ((a v c) ^ (b v c))
51comcom2 183 . . . . . 6 b C c'
65, 3fh4rc 482 . . . . 5 ((a ^ b) v c') = ((a v c') ^ (b v c'))
74, 62an 79 . . . 4 (((a ^ b) v c) ^ ((a ^ b) v c')) = (((a v c) ^ (b v c)) ^ ((a v c') ^ (b v c')))
8 an4 86 . . . 4 (((a v c) ^ (b v c)) ^ ((a v c') ^ (b v c'))) = (((a v c) ^ (a v c')) ^ ((b v c) ^ (b v c')))
9 an32 83 . . . . 5 (((a v c) ^ (a v c')) ^ b) = (((a v c) ^ b) ^ (a v c'))
101comd 456 . . . . . 6 b = ((b v c) ^ (b v c'))
1110lan 77 . . . . 5 (((a v c) ^ (a v c')) ^ b) = (((a v c) ^ (a v c')) ^ ((b v c) ^ (b v c')))
123, 1fh1r 473 . . . . . . 7 ((a v c) ^ b) = ((a ^ b) v (c ^ b))
1312ran 78 . . . . . 6 (((a v c) ^ b) ^ (a v c')) = (((a ^ b) v (c ^ b)) ^ (a v c'))
14 lea 160 . . . . . . . . . 10 (a ^ b) =< a
15 leo 158 . . . . . . . . . 10 a =< (a v c')
1614, 15letr 137 . . . . . . . . 9 (a ^ b) =< (a v c')
1716lecom 180 . . . . . . . 8 (a ^ b) C (a v c')
1817comcom 453 . . . . . . 7 (a v c') C (a ^ b)
19 gsth.3 . . . . . . . . . . 11 a C (b ^ c)
2019comcom 453 . . . . . . . . . 10 (b ^ c) C a
21 coman2 186 . . . . . . . . . . 11 (b ^ c) C c
2221comcom2 183 . . . . . . . . . 10 (b ^ c) C c'
2320, 22com2or 483 . . . . . . . . 9 (b ^ c) C (a v c')
2423comcom 453 . . . . . . . 8 (a v c') C (b ^ c)
25 ancom 74 . . . . . . . 8 (b ^ c) = (c ^ b)
2624, 25cbtr 182 . . . . . . 7 (a v c') C (c ^ b)
2718, 26fh1r 473 . . . . . 6 (((a ^ b) v (c ^ b)) ^ (a v c')) = (((a ^ b) ^ (a v c')) v ((c ^ b) ^ (a v c')))
2816df2le2 136 . . . . . . . 8 ((a ^ b) ^ (a v c')) = (a ^ b)
29 ancom 74 . . . . . . . . . 10 (c ^ b) = (b ^ c)
3029ran 78 . . . . . . . . 9 ((c ^ b) ^ (a v c')) = ((b ^ c) ^ (a v c'))
3120, 22fh1 469 . . . . . . . . 9 ((b ^ c) ^ (a v c')) = (((b ^ c) ^ a) v ((b ^ c) ^ c'))
32 anass 76 . . . . . . . . . . . 12 ((b ^ c) ^ c') = (b ^ (c ^ c'))
33 dff 101 . . . . . . . . . . . . . 14 0 = (c ^ c')
3433ax-r1 35 . . . . . . . . . . . . 13 (c ^ c') = 0
3534lan 77 . . . . . . . . . . . 12 (b ^ (c ^ c')) = (b ^ 0)
36 an0 108 . . . . . . . . . . . 12 (b ^ 0) = 0
3732, 35, 363tr 65 . . . . . . . . . . 11 ((b ^ c) ^ c') = 0
3837lor 70 . . . . . . . . . 10 (((b ^ c) ^ a) v ((b ^ c) ^ c')) = (((b ^ c) ^ a) v 0)
39 or0 102 . . . . . . . . . 10 (((b ^ c) ^ a) v 0) = ((b ^ c) ^ a)
4038, 39ax-r2 36 . . . . . . . . 9 (((b ^ c) ^ a) v ((b ^ c) ^ c')) = ((b ^ c) ^ a)
4130, 31, 403tr 65 . . . . . . . 8 ((c ^ b) ^ (a v c')) = ((b ^ c) ^ a)
4228, 412or 72 . . . . . . 7 (((a ^ b) ^ (a v c')) v ((c ^ b) ^ (a v c'))) = ((a ^ b) v ((b ^ c) ^ a))
43 ax-a2 31 . . . . . . 7 ((a ^ b) v ((b ^ c) ^ a)) = (((b ^ c) ^ a) v (a ^ b))
44 ancom 74 . . . . . . . . 9 ((b ^ c) ^ a) = (a ^ (b ^ c))
45 lea 160 . . . . . . . . . 10 (b ^ c) =< b
4645lelan 167 . . . . . . . . 9 (a ^ (b ^ c)) =< (a ^ b)
4744, 46bltr 138 . . . . . . . 8 ((b ^ c) ^ a) =< (a ^ b)
4847df-le2 131 . . . . . . 7 (((b ^ c) ^ a) v (a ^ b)) = (a ^ b)
4942, 43, 483tr 65 . . . . . 6 (((a ^ b) ^ (a v c')) v ((c ^ b) ^ (a v c'))) = (a ^ b)
5013, 27, 493tr 65 . . . . 5 (((a v c) ^ b) ^ (a v c')) = (a ^ b)
519, 11, 503tr2 64 . . . 4 (((a v c) ^ (a v c')) ^ ((b v c) ^ (b v c'))) = (a ^ b)
527, 8, 513tr 65 . . 3 (((a ^ b) v c) ^ ((a ^ b) v c')) = (a ^ b)
5352ax-r1 35 . 2 (a ^ b) = (((a ^ b) v c) ^ ((a ^ b) v c'))
5453df2c1 468 1 (a ^ b) C c
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7  0wf 9
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  gsth2  490
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