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Theorem mdandysum2p2e4 27944
Description: CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half-adders and full adders in propositional calculus

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

Hypotheses
Ref Expression
mdandysum2p2e4.1  |-  (jth  <->  F.  )
mdandysum2p2e4.2  |-  (jta  <->  T.  )
mdandysum2p2e4.a  |-  ( ph  <->  ( th  /\  ta )
)
mdandysum2p2e4.b  |-  ( ps  <->  ( et  /\  ze )
)
mdandysum2p2e4.c  |-  ( ch  <->  ( si  /\  rh ) )
mdandysum2p2e4.d  |-  ( th  <-> jth
)
mdandysum2p2e4.e  |-  ( ta  <-> jth
)
mdandysum2p2e4.f  |-  ( et  <-> jta
)
mdandysum2p2e4.g  |-  ( ze  <-> jta
)
mdandysum2p2e4.h  |-  ( si  <-> jth
)
mdandysum2p2e4.i  |-  ( rh  <-> jth
)
mdandysum2p2e4.j  |-  ( mu  <-> jth
)
mdandysum2p2e4.k  |-  ( la  <-> jth
)
mdandysum2p2e4.l  |-  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) )
mdandysum2p2e4.m  |-  (jph  <->  ( ( et \/_ ze )  \/  ph ) )
mdandysum2p2e4.n  |-  (jps  <->  ( ( si \/_ rh )  \/  ps )
)
mdandysum2p2e4.o  |-  (jch  <->  ( ( mu \/_ la )  \/  ch ) )
Assertion
Ref Expression
mdandysum2p2e4  |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph  <->  ( th  /\  ta ) )  /\  ( ps  <->  ( et  /\  ze ) ) )  /\  ( ch  <->  ( si  /\  rh ) ) )  /\  ( th  <->  F.  ) )  /\  ( ta  <->  F.  )
)  /\  ( et  <->  T.  ) )  /\  ( ze 
<->  T.  ) )  /\  ( si  <->  F.  ) )  /\  ( rh  <->  F.  )
)  /\  ( mu  <->  F.  ) )  /\  ( la 
<->  F.  ) )  /\  ( ka  <->  ( ( th
\/_ ta ) \/_ ( th  /\  ta ) ) ) )  /\  (jph  <->  ( ( et \/_ ze )  \/  ph ) ) )  /\  (jps  <->  ( ( si \/_ rh )  \/  ps )
) )  /\  (jch  <->  ( ( mu \/_ la )  \/  ch ) ) )  ->  ( ( ( ( ka  <->  F.  )  /\  (jph  <->  F.  )
)  /\  (jps  <->  T.  ) )  /\  (jch  <->  F.  ) ) )

Proof of Theorem mdandysum2p2e4
StepHypRef Expression
1 mdandysum2p2e4.a . 2  |-  ( ph  <->  ( th  /\  ta )
)
2 mdandysum2p2e4.b . 2  |-  ( ps  <->  ( et  /\  ze )
)
3 mdandysum2p2e4.c . 2  |-  ( ch  <->  ( si  /\  rh ) )
4 mdandysum2p2e4.d . . 3  |-  ( th  <-> jth
)
5 mdandysum2p2e4.1 . . 3  |-  (jth  <->  F.  )
64, 5aisbbisfaisf 27870 . 2  |-  ( th  <->  F.  )
7 mdandysum2p2e4.e . . 3  |-  ( ta  <-> jth
)
87, 5aisbbisfaisf 27870 . 2  |-  ( ta  <->  F.  )
9 mdandysum2p2e4.f . . 3  |-  ( et  <-> jta
)
10 mdandysum2p2e4.2 . . 3  |-  (jta  <->  T.  )
119, 10aiffbbtat 27869 . 2  |-  ( et  <->  T.  )
12 mdandysum2p2e4.g . . 3  |-  ( ze  <-> jta
)
1312, 10aiffbbtat 27869 . 2  |-  ( ze  <->  T.  )
14 mdandysum2p2e4.h . . 3  |-  ( si  <-> jth
)
1514, 5aisbbisfaisf 27870 . 2  |-  ( si  <->  F.  )
16 mdandysum2p2e4.i . . 3  |-  ( rh  <-> jth
)
1716, 5aisbbisfaisf 27870 . 2  |-  ( rh  <->  F.  )
18 mdandysum2p2e4.j . . 3  |-  ( mu  <-> jth
)
1918, 5aisbbisfaisf 27870 . 2  |-  ( mu  <->  F.  )
20 mdandysum2p2e4.k . . 3  |-  ( la  <-> jth
)
2120, 5aisbbisfaisf 27870 . 2  |-  ( la  <->  F.  )
22 mdandysum2p2e4.l . 2  |-  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) )
23 mdandysum2p2e4.m . 2  |-  (jph  <->  ( ( et \/_ ze )  \/  ph ) )
24 mdandysum2p2e4.n . 2  |-  (jps  <->  ( ( si \/_ rh )  \/  ps )
)
25 mdandysum2p2e4.o . 2  |-  (jch  <->  ( ( mu \/_ la )  \/  ch ) )
261, 2, 3, 6, 8, 11, 13, 15, 17, 19, 21, 22, 23, 24, 25dandysum2p2e4 27943 1  |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph  <->  ( th  /\  ta ) )  /\  ( ps  <->  ( et  /\  ze ) ) )  /\  ( ch  <->  ( si  /\  rh ) ) )  /\  ( th  <->  F.  ) )  /\  ( ta  <->  F.  )
)  /\  ( et  <->  T.  ) )  /\  ( ze 
<->  T.  ) )  /\  ( si  <->  F.  ) )  /\  ( rh  <->  F.  )
)  /\  ( mu  <->  F.  ) )  /\  ( la 
<->  F.  ) )  /\  ( ka  <->  ( ( th
\/_ ta ) \/_ ( th  /\  ta ) ) ) )  /\  (jph  <->  ( ( et \/_ ze )  \/  ph ) ) )  /\  (jps  <->  ( ( si \/_ rh )  \/  ps )
) )  /\  (jch  <->  ( ( mu \/_ la )  \/  ch ) ) )  ->  ( ( ( ( ka  <->  F.  )  /\  (jph  <->  F.  )
)  /\  (jps  <->  T.  ) )  /\  (jch  <->  F.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   \/_wxo 1295    T. wtru 1307    F. wfal 1308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-xor 1296  df-tru 1310  df-fal 1311
  Copyright terms: Public domain W3C validator