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Theorem staddi 22828
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 22798 . . . . . . 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 22798 . . . . . . 7  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
64, 5mpi 16 . . . . . 6  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
73, 6readdcld 8864 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  e.  RR )
8 ltne 8919 . . . . . 6  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  2  =/=  (
( S `  A
)  +  ( S `
 B ) ) )
98necomd 2531 . . . . 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 2497 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  -.  (
( S `  A
)  +  ( S `
 B ) )  <  2 ) )
13 1re 8839 . . . . . . . . 9  |-  1  e.  RR
1413a1i 10 . . . . . . . 8  |-  ( S  e.  States  ->  1  e.  RR )
15 stle1 22807 . . . . . . . . 9  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
164, 15mpi 16 . . . . . . . 8  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
176, 14, 3, 16leadd2dd 9389 . . . . . . 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 9243 . . . . . . . . 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 8822 . . . . . . . . 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 8852 . . . . . . . . 9  |-  ( 1  +  1 )  e.  RR
2625a1i 10 . . . . . . . 8  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
27 lelttr 8914 . . . . . . . 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 9806 . . . . 5  |-  2  =  ( 1  +  1 )
3230, 31syl6breqr 4065 . . . 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 22807 . . . . 5  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
361, 35mpi 16 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
37 leloe 8910 . . . . 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 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   RRcr 8738   1c1 8740    + caddc 8742    < clt 8869    <_ cle 8870   2c2 9797   CHcch 21511   Statescst 21544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-hilex 21581
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-2 9806  df-icc 10665  df-sh 21788  df-ch 21803  df-st 22793
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