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Theorem staddi 22826
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 22796 . . . . . . 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 22796 . . . . . . 7  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
64, 5mpi 16 . . . . . 6  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
73, 6readdcld 8862 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( S `  B
) )  e.  RR )
8 ltne 8917 . . . . . 6  |-  ( ( ( ( S `  A )  +  ( S `  B ) )  e.  RR  /\  ( ( S `  A )  +  ( S `  B ) )  <  2 )  ->  2  =/=  (
( S `  A
)  +  ( S `
 B ) ) )
98necomd 2529 . . . . 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 2495 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  =  2  ->  -.  (
( S `  A
)  +  ( S `
 B ) )  <  2 ) )
13 1re 8837 . . . . . . . . 9  |-  1  e.  RR
1413a1i 10 . . . . . . . 8  |-  ( S  e.  States  ->  1  e.  RR )
15 stle1 22805 . . . . . . . . 9  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
164, 15mpi 16 . . . . . . . 8  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
176, 14, 3, 16leadd2dd 9387 . . . . . . 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 9241 . . . . . . . . 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 8820 . . . . . . . . 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 8850 . . . . . . . . 9  |-  ( 1  +  1 )  e.  RR
2625a1i 10 . . . . . . . 8  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
27 lelttr 8912 . . . . . . . 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 9804 . . . . 5  |-  2  =  ( 1  +  1 )
3230, 31syl6breqr 4063 . . . 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 22805 . . . . 5  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
361, 35mpi 16 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
37 leloe 8908 . . . . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868   2c2 9795   CHcch 21509   Statescst 21542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hilex 21579
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-2 9804  df-icc 10663  df-sh 21786  df-ch 21801  df-st 22791
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