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Theorem staddi 26829
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 26799 . . . . . . 7  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 17 . . . . . 6  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
4 stle.2 . . . . . . 7  |-  B  e. 
CH
5 stcl 26799 . . . . . . 7  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
64, 5mpi 17 . . . . . 6  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
73, 6readdcld 9614 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  e.  RR )
8 ltne 9672 . . . . . 6  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  2  =/=  (
( S `  A
)  +  ( S `
 B ) ) )
98necomd 2733 . . . . 5  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  ( ( S `
 A )  +  ( S `  B
) )  =/=  2
)
107, 9sylan 471 . . . 4  |-  ( ( S  e.  States  /\  (
( S `  A
)  +  ( S `
 B ) )  <  2 )  -> 
( ( S `  A )  +  ( S `  B ) )  =/=  2 )
1110ex 434 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  <  2  ->  ( ( S `  A )  +  ( S `  B ) )  =/=  2 ) )
1211necon2bd 2677 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  -.  (
( S `  A
)  +  ( S `
 B ) )  <  2 ) )
13 1re 9586 . . . . . . . . 9  |-  1  e.  RR
1413a1i 11 . . . . . . . 8  |-  ( S  e.  States  ->  1  e.  RR )
15 stle1 26808 . . . . . . . . 9  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
164, 15mpi 17 . . . . . . . 8  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
176, 14, 3, 16leadd2dd 10158 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  <_  (
( S `  A
)  +  1 ) )
1817adantr 465 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 ) )
19 ltadd1 10010 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  1  e.  RR )  ->  (
( S `  A
)  <  1  <->  ( ( S `  A )  +  1 )  < 
( 1  +  1 ) ) )
2019biimpd 207 . . . . . . . 8  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  1  e.  RR )  ->  (
( S `  A
)  <  1  ->  ( ( S `  A
)  +  1 )  <  ( 1  +  1 ) ) )
213, 14, 14, 20syl3anc 1223 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  1 )  < 
( 1  +  1 ) ) )
2221imp 429 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )
23 readdcl 9566 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  +  1 )  e.  RR )
243, 13, 23sylancl 662 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  1 )  e.  RR )
2513, 13readdcli 9600 . . . . . . . . 9  |-  ( 1  +  1 )  e.  RR
2625a1i 11 . . . . . . . 8  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
27 lelttr 9666 . . . . . . . 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 1223 . . . . . . 7  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 )  /\  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )  -> 
( ( S `  A )  +  ( S `  B ) )  <  ( 1  +  1 ) ) )
2928adantr 465 . . . . . 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 679 . . . . 5  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <  ( 1  +  1 ) )
31 df-2 10585 . . . . 5  |-  2  =  ( 1  +  1 )
3230, 31syl6breqr 4482 . . . 4  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <  2 )
3332ex 434 . . 3  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( S `  B ) )  <  2 ) )
3433con3d 133 . 2  |-  ( S  e.  States  ->  ( -.  (
( S `  A
)  +  ( S `
 B ) )  <  2  ->  -.  ( S `  A )  <  1 ) )
35 stle1 26808 . . . . 5  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
361, 35mpi 17 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
37 leloe 9662 . . . . 5  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
383, 13, 37sylancl 662 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <_ 
1  <->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) ) )
3936, 38mpbid 210 . . 3  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) )
4039ord 377 . 2  |-  ( S  e.  States  ->  ( -.  ( S `  A )  <  1  ->  ( S `  A )  =  1 ) )
4112, 34, 403syld 55 1  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  ( S `  A )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   RRcr 9482   1c1 9484    + caddc 9486    < clt 9619    <_ cle 9620   2c2 10576   CHcch 25510   Statescst 25543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-hilex 25580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-2 10585  df-icc 11527  df-sh 25788  df-ch 25803  df-st 26794
This theorem is referenced by: (None)
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