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Theorem sa5 836
Description: Possible axiom for a "Sasaki algebra" for orthoarguesian lattices.
Hypothesis
Ref Expression
sa5.1 (a ->1 c) =< (b ->1 c)
Assertion
Ref Expression
sa5 (b' ->1 c) =< ((a' ->1 c) v c)
Proof of Theorem sa5
StepHypRef Expression
1 leor 159 . . . . 5 b =< (c v b)
2 ax-a2 31 . . . . . . . . . 10 (b' v c') = (c' v b')
32lan 77 . . . . . . . . 9 (b ^ (b' v c')) = (b ^ (c' v b'))
43ax-r5 38 . . . . . . . 8 ((b ^ (b' v c')) v c) = ((b ^ (c' v b')) v c)
5 ax-a2 31 . . . . . . . 8 ((b ^ (c' v b')) v c) = (c v (b ^ (c' v b')))
6 oml6 488 . . . . . . . 8 (c v (b ^ (c' v b'))) = (c v b)
74, 5, 63tr 65 . . . . . . 7 ((b ^ (b' v c')) v c) = (c v b)
87ax-r1 35 . . . . . 6 (c v b) = ((b ^ (b' v c')) v c)
9 sa5.1 . . . . . . . . . 10 (a ->1 c) =< (b ->1 c)
109lecon 154 . . . . . . . . 9 (b ->1 c)' =< (a ->1 c)'
11 ud1lem0c 277 . . . . . . . . 9 (b ->1 c)' = (b ^ (b' v c'))
12 ud1lem0c 277 . . . . . . . . 9 (a ->1 c)' = (a ^ (a' v c'))
1310, 11, 12le3tr2 141 . . . . . . . 8 (b ^ (b' v c')) =< (a ^ (a' v c'))
14 lea 160 . . . . . . . 8 (a ^ (a' v c')) =< a
1513, 14letr 137 . . . . . . 7 (b ^ (b' v c')) =< a
1615leror 152 . . . . . 6 ((b ^ (b' v c')) v c) =< (a v c)
178, 16bltr 138 . . . . 5 (c v b) =< (a v c)
181, 17letr 137 . . . 4 b =< (a v c)
19 ax-a1 30 . . . 4 b = b''
20 ax-a1 30 . . . . . 6 a = a''
21 ax-a2 31 . . . . . . 7 (c v (c ^ a')) = ((c ^ a') v c)
22 orabs 120 . . . . . . 7 (c v (c ^ a')) = c
23 ancom 74 . . . . . . . 8 (c ^ a') = (a' ^ c)
2423ax-r5 38 . . . . . . 7 ((c ^ a') v c) = ((a' ^ c) v c)
2521, 22, 243tr2 64 . . . . . 6 c = ((a' ^ c) v c)
2620, 252or 72 . . . . 5 (a v c) = (a'' v ((a' ^ c) v c))
27 ax-a3 32 . . . . . 6 ((a'' v (a' ^ c)) v c) = (a'' v ((a' ^ c) v c))
2827ax-r1 35 . . . . 5 (a'' v ((a' ^ c) v c)) = ((a'' v (a' ^ c)) v c)
2926, 28ax-r2 36 . . . 4 (a v c) = ((a'' v (a' ^ c)) v c)
3018, 19, 29le3tr2 141 . . 3 b'' =< ((a'' v (a' ^ c)) v c)
31 lear 161 . . . 4 (b' ^ c) =< c
32 leor 159 . . . 4 c =< ((a'' v (a' ^ c)) v c)
3331, 32letr 137 . . 3 (b' ^ c) =< ((a'' v (a' ^ c)) v c)
3430, 33lel2or 170 . 2 (b'' v (b' ^ c)) =< ((a'' v (a' ^ c)) v c)
35 df-i1 44 . 2 (b' ->1 c) = (b'' v (b' ^ c))
36 df-i1 44 . . 3 (a' ->1 c) = (a'' v (a' ^ c))
3736ax-r5 38 . 2 ((a' ->1 c) v c) = ((a'' v (a' ^ c)) v c)
3834, 35, 37le3tr1 140 1 (b' ->1 c) =< ((a' ->1 c) v c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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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