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Theorem staddi 23599
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 23569 . . . . . . 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 23569 . . . . . . 7  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
64, 5mpi 17 . . . . . 6  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
73, 6readdcld 9050 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  e.  RR )
8 ltne 9105 . . . . . 6  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  2  =/=  (
( S `  A
)  +  ( S `
 B ) ) )
98necomd 2635 . . . . 5  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  ( ( S `
 A )  +  ( S `  B
) )  =/=  2
)
107, 9sylan 458 . . . 4  |-  ( ( S  e.  States  /\  (
( S `  A
)  +  ( S `
 B ) )  <  2 )  -> 
( ( S `  A )  +  ( S `  B ) )  =/=  2 )
1110ex 424 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  <  2  ->  ( ( S `  A )  +  ( S `  B ) )  =/=  2 ) )
1211necon2bd 2601 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  -.  (
( S `  A
)  +  ( S `
 B ) )  <  2 ) )
13 1re 9025 . . . . . . . . 9  |-  1  e.  RR
1413a1i 11 . . . . . . . 8  |-  ( S  e.  States  ->  1  e.  RR )
15 stle1 23578 . . . . . . . . 9  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
164, 15mpi 17 . . . . . . . 8  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
176, 14, 3, 16leadd2dd 9575 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  <_  (
( S `  A
)  +  1 ) )
1817adantr 452 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 ) )
19 ltadd1 9429 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  1  e.  RR )  ->  (
( S `  A
)  <  1  <->  ( ( S `  A )  +  1 )  < 
( 1  +  1 ) ) )
2019biimpd 199 . . . . . . . 8  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  1  e.  RR )  ->  (
( S `  A
)  <  1  ->  ( ( S `  A
)  +  1 )  <  ( 1  +  1 ) ) )
213, 14, 14, 20syl3anc 1184 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  1 )  < 
( 1  +  1 ) ) )
2221imp 419 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )
23 readdcl 9008 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  +  1 )  e.  RR )
243, 13, 23sylancl 644 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  1 )  e.  RR )
2513, 13readdcli 9038 . . . . . . . . 9  |-  ( 1  +  1 )  e.  RR
2625a1i 11 . . . . . . . 8  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
27 lelttr 9100 . . . . . . . 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 1184 . . . . . . 7  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  <_  ( ( S `
 A )  +  1 )  /\  (
( S `  A
)  +  1 )  <  ( 1  +  1 ) )  -> 
( ( S `  A )  +  ( S `  B ) )  <  ( 1  +  1 ) ) )
2928adantr 452 . . . . . 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 661 . . . . 5  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <  ( 1  +  1 ) )
31 df-2 9992 . . . . 5  |-  2  =  ( 1  +  1 )
3230, 31syl6breqr 4195 . . . 4  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( S `
 B ) )  <  2 )
3332ex 424 . . 3  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( S `  B ) )  <  2 ) )
3433con3d 127 . 2  |-  ( S  e.  States  ->  ( -.  (
( S `  A
)  +  ( S `
 B ) )  <  2  ->  -.  ( S `  A )  <  1 ) )
35 stle1 23578 . . . . 5  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
361, 35mpi 17 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
37 leloe 9096 . . . . 5  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
383, 13, 37sylancl 644 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <_ 
1  <->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) ) )
3936, 38mpbid 202 . . 3  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) )
4039ord 367 . 2  |-  ( S  e.  States  ->  ( -.  ( S `  A )  <  1  ->  ( S `  A )  =  1 ) )
4112, 34, 403syld 53 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   RRcr 8924   1c1 8926    + caddc 8928    < clt 9055    <_ cle 9056   2c2 9983   CHcch 22282   Statescst 22315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-hilex 22352
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-2 9992  df-icc 10857  df-sh 22559  df-ch 22574  df-st 23564
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