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Theorem stadd3i 27736
Description: If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
stle.1  |-  A  e. 
CH
stle.2  |-  B  e. 
CH
stm1add3.3  |-  C  e. 
CH
Assertion
Ref Expression
stadd3i  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  ->  ( S `  A )  =  1 ) )

Proof of Theorem stadd3i
StepHypRef Expression
1 stle.1 . . . . . 6  |-  A  e. 
CH
2 stcl 27704 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 21 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
43recnd 9668 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  e.  CC )
5 stle.2 . . . . . 6  |-  B  e. 
CH
6 stcl 27704 . . . . . 6  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
75, 6mpi 21 . . . . 5  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
87recnd 9668 . . . 4  |-  ( S  e.  States  ->  ( S `  B )  e.  CC )
9 stm1add3.3 . . . . . 6  |-  C  e. 
CH
10 stcl 27704 . . . . . 6  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  e.  RR ) )
119, 10mpi 21 . . . . 5  |-  ( S  e.  States  ->  ( S `  C )  e.  RR )
1211recnd 9668 . . . 4  |-  ( S  e.  States  ->  ( S `  C )  e.  CC )
134, 8, 12addassd 9664 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  +  ( S `  C
) )  =  ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) ) )
1413eqeq1d 2431 . 2  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  <->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3 ) )
15 eqcom 2438 . . . 4  |-  ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3  <->  3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) )
167, 11readdcld 9669 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  e.  RR )
173, 16readdcld 9669 . . . . . 6  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  e.  RR )
18 3re 10683 . . . . . 6  |-  3  e.  RR
19 ltneOLD 9730 . . . . . . 7  |-  ( ( ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  e.  RR  /\  3  e.  RR  /\  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 )  -> 
3  =/=  ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) ) )
20193exp 1204 . . . . . 6  |-  ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  e.  RR  ->  (
3  e.  RR  ->  ( ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  <  3  -> 
3  =/=  ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) ) ) ) )
2117, 18, 20mpisyl 22 . . . . 5  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  <  3  ->  3  =/=  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) ) )
2221necon2bd 2646 . . . 4  |-  ( S  e.  States  ->  ( 3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  ->  -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 ) )
2315, 22syl5bi 220 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  =  3  ->  -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 ) )
24 1re 9641 . . . . . . . . . . 11  |-  1  e.  RR
2524, 24readdcli 9655 . . . . . . . . . 10  |-  ( 1  +  1 )  e.  RR
2625a1i 11 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
2724a1i 11 . . . . . . . . . 10  |-  ( S  e.  States  ->  1  e.  RR )
28 stle1 27713 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
295, 28mpi 21 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
30 stle1 27713 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  <_  1
) )
319, 30mpi 21 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  C )  <_  1
)
327, 11, 27, 27, 29, 31le2addd 10231 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  <_  (
1  +  1 ) )
3316, 26, 3, 32leadd2dd 10227 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  (
( S `  A
)  +  ( 1  +  1 ) ) )
3433adantr 466 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  ( ( S `
 A )  +  ( 1  +  1 ) ) )
35 ltadd1 10080 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  (
1  +  1 )  e.  RR )  -> 
( ( S `  A )  <  1  <->  ( ( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) ) )
3635biimpd 210 . . . . . . . . 9  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  (
1  +  1 )  e.  RR )  -> 
( ( S `  A )  <  1  ->  ( ( S `  A )  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) ) )
373, 27, 26, 36syl3anc 1264 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( 1  +  1 ) )  < 
( 1  +  ( 1  +  1 ) ) ) )
3837imp 430 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) )
39 readdcl 9621 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  ( 1  +  1 )  e.  RR )  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
403, 25, 39sylancl 666 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
4124, 25readdcli 9655 . . . . . . . . . 10  |-  ( 1  +  ( 1  +  1 ) )  e.  RR
4241a1i 11 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  ( 1  +  1 ) )  e.  RR )
43 lelttr 9723 . . . . . . . . 9  |-  ( ( ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  e.  RR  /\  ( ( S `  A )  +  ( 1  +  1 ) )  e.  RR  /\  ( 1  +  ( 1  +  1 ) )  e.  RR )  ->  ( ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  ( ( S `
 A )  +  ( 1  +  1 ) )  /\  (
( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) )  -> 
( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  <  ( 1  +  ( 1  +  1 ) ) ) )
4417, 40, 42, 43syl3anc 1264 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  ( ( S `
 A )  +  ( 1  +  1 ) )  /\  (
( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) )  -> 
( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  <  ( 1  +  ( 1  +  1 ) ) ) )
4544adantr 466 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  (
( S `  A
)  +  ( 1  +  1 ) )  /\  ( ( S `
 A )  +  ( 1  +  1 ) )  <  (
1  +  ( 1  +  1 ) ) )  ->  ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  < 
( 1  +  ( 1  +  1 ) ) ) )
4634, 38, 45mp2and 683 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  ( 1  +  ( 1  +  1 ) ) )
47 df-3 10669 . . . . . . 7  |-  3  =  ( 2  +  1 )
48 df-2 10668 . . . . . . . 8  |-  2  =  ( 1  +  1 )
4948oveq1i 6315 . . . . . . 7  |-  ( 2  +  1 )  =  ( ( 1  +  1 )  +  1 )
50 ax-1cn 9596 . . . . . . . 8  |-  1  e.  CC
5150, 50, 50addassi 9650 . . . . . . 7  |-  ( ( 1  +  1 )  +  1 )  =  ( 1  +  ( 1  +  1 ) )
5247, 49, 513eqtrri 2463 . . . . . 6  |-  ( 1  +  ( 1  +  1 ) )  =  3
5346, 52syl6breq 4465 . . . . 5  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 )
5453ex 435 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  <  3 ) )
5554con3d 138 . . 3  |-  ( S  e.  States  ->  ( -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3  ->  -.  ( S `  A )  <  1 ) )
56 stle1 27713 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
571, 56mpi 21 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
58 leloe 9719 . . . . . 6  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
593, 24, 58sylancl 666 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  <_ 
1  <->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) ) )
6057, 59mpbid 213 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) )
6160ord 378 . . 3  |-  ( S  e.  States  ->  ( -.  ( S `  A )  <  1  ->  ( S `  A )  =  1 ) )
6223, 55, 613syld 57 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  =  3  ->  ( S `  A )  =  1 ) )
6314, 62sylbid 218 1  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  ->  ( S `  A )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   RRcr 9537   1c1 9539    + caddc 9541    < clt 9674    <_ cle 9675   2c2 10659   3c3 10660   CHcch 26417   Statescst 26450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-hilex 26487
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-2 10668  df-3 10669  df-icc 11642  df-sh 26695  df-ch 26709  df-st 27699
This theorem is referenced by:  golem2  27760
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