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Theorem staddi 22842
Description: If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
stle.1  |-  A  e. 
CH
stle.2  |-  B  e. 
CH
Assertion
Ref Expression
staddi  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  ( S `  A )  =  1 ) )

Proof of Theorem staddi
StepHypRef Expression
1 stle.1 . . . . . . 7  |-  A  e. 
CH
2 stcl 22812 . . . . . . 7  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 16 . . . . . 6  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
4 stle.2 . . . . . . 7  |-  B  e. 
CH
5 stcl 22812 . . . . . . 7  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
64, 5mpi 16 . . . . . 6  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
73, 6readdcld 8878 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  e.  RR )
8 ltne 8933 . . . . . 6  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  2  =/=  (
( S `  A
)  +  ( S `
 B ) ) )
98necomd 2542 . . . . 5  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  ( ( S `
 A )  +  ( S `  B
) )  =/=  2
)
107, 9sylan 457 . . . 4  |-  ( ( S  e.  States  /\  (
( S `  A
)  +  ( S `
 B ) )  <  2 )  -> 
( ( S `  A )  +  ( S `  B ) )  =/=  2 )
1110ex 423 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  <  2  ->  ( ( S `  A )  +  ( S `  B ) )  =/=  2 ) )
1211necon2bd 2508 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  -.  (
( S `  A
)  +  ( S `
 B ) )  <  2 ) )
13 1re 8853 . . . . . . . . 9  |-  1  e.  RR
1413a1i 10 . . . . . . . 8  |-  ( S  e.  States  ->  1  e.  RR )
15 stle1 22821 . . . . . . . . 9  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
164, 15mpi 16 . . . . . . . 8  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
176, 14, 3, 16leadd2dd 9403 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  <_  (
( S `  A
)  +  1 ) )
1817adantr 451 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 ) )
19 ltadd1 9257 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  1  e.  RR )  ->  (
( S `  A
)  <  1  <->  ( ( S `  A )  +  1 )  < 
( 1  +  1 ) ) )
2019biimpd 198 . . . . . . . 8  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  1  e.  RR )  ->  (
( S `  A
)  <  1  ->  ( ( S `  A
)  +  1 )  <  ( 1  +  1 ) ) )
213, 14, 14, 20syl3anc 1182 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  1 )  < 
( 1  +  1 ) ) )
2221imp 418 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )
23 readdcl 8836 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  +  1 )  e.  RR )
243, 13, 23sylancl 643 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  1 )  e.  RR )
2513, 13readdcli 8866 . . . . . . . . 9  |-  ( 1  +  1 )  e.  RR
2625a1i 10 . . . . . . . 8  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
27 lelttr 8928 . . . . . . . 8  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  1 )  e.  RR  /\  ( 1  +  1 )  e.  RR )  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 )  /\  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )  -> 
( ( S `  A )  +  ( S `  B ) )  <  ( 1  +  1 ) ) )
287, 24, 26, 27syl3anc 1182 . . . . . . 7  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 )  /\  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )  -> 
( ( S `  A )  +  ( S `  B ) )  <  ( 1  +  1 ) ) )
2928adantr 451 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( ( ( S `
 A )  +  ( S `  B
) )  <_  (
( S `  A
)  +  1 )  /\  ( ( S `
 A )  +  1 )  <  (
1  +  1 ) )  ->  ( ( S `  A )  +  ( S `  B ) )  < 
( 1  +  1 ) ) )
3018, 22, 29mp2and 660 . . . . 5  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <  ( 1  +  1 ) )
31 df-2 9820 . . . . 5  |-  2  =  ( 1  +  1 )
3230, 31syl6breqr 4079 . . . 4  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <  2 )
3332ex 423 . . 3  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( S `  B ) )  <  2 ) )
3433con3d 125 . 2  |-  ( S  e.  States  ->  ( -.  (
( S `  A
)  +  ( S `
 B ) )  <  2  ->  -.  ( S `  A )  <  1 ) )
35 stle1 22821 . . . . 5  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
361, 35mpi 16 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
37 leloe 8924 . . . . 5  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
383, 13, 37sylancl 643 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <_ 
1  <->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) ) )
3936, 38mpbid 201 . . 3  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) )
4039ord 366 . 2  |-  ( S  e.  States  ->  ( -.  ( S `  A )  <  1  ->  ( S `  A )  =  1 ) )
4112, 34, 403syld 51 1  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  ( S `  A )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884   2c2 9811   CHcch 21525   Statescst 21558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-2 9820  df-icc 10679  df-sh 21802  df-ch 21817  df-st 22807
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