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Theorem stadd3i 23141
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 23109 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 16 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
43recnd 9008 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  e.  CC )
5 stle.2 . . . . . 6  |-  B  e. 
CH
6 stcl 23109 . . . . . 6  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
75, 6mpi 16 . . . . 5  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
87recnd 9008 . . . 4  |-  ( S  e.  States  ->  ( S `  B )  e.  CC )
9 stm1add3.3 . . . . . 6  |-  C  e. 
CH
10 stcl 23109 . . . . . 6  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  e.  RR ) )
119, 10mpi 16 . . . . 5  |-  ( S  e.  States  ->  ( S `  C )  e.  RR )
1211recnd 9008 . . . 4  |-  ( S  e.  States  ->  ( S `  C )  e.  CC )
134, 8, 12addassd 9004 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  +  ( S `  C
) )  =  ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) ) )
1413eqeq1d 2374 . 2  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  <->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3 ) )
15 eqcom 2368 . . . 4  |-  ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3  <->  3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) )
167, 11readdcld 9009 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  e.  RR )
173, 16readdcld 9009 . . . . . 6  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  e.  RR )
18 3re 9964 . . . . . 6  |-  3  e.  RR
19 ltneOLD 9065 . . . . . . 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 1151 . . . . . 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, 20ee10 1381 . . . . 5  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  <  3  ->  3  =/=  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) ) )
2221necon2bd 2578 . . . 4  |-  ( S  e.  States  ->  ( 3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  ->  -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 ) )
2315, 22syl5bi 208 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  =  3  ->  -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 ) )
24 1re 8984 . . . . . . . . . . 11  |-  1  e.  RR
2524, 24readdcli 8997 . . . . . . . . . 10  |-  ( 1  +  1 )  e.  RR
2625a1i 10 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  1 )  e.  RR )
2724a1i 10 . . . . . . . . . 10  |-  ( S  e.  States  ->  1  e.  RR )
28 stle1 23118 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
295, 28mpi 16 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
30 stle1 23118 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  <_  1
) )
319, 30mpi 16 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  C )  <_  1
)
327, 11, 27, 27, 29, 31le2addd 9537 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  <_  (
1  +  1 ) )
3316, 26, 3, 32leadd2dd 9534 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  (
( S `  A
)  +  ( 1  +  1 ) ) )
3433adantr 451 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  ( ( S `
 A )  +  ( 1  +  1 ) ) )
35 ltadd1 9388 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  (
1  +  1 )  e.  RR )  -> 
( ( S `  A )  <  1  <->  ( ( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) ) )
3635biimpd 198 . . . . . . . . 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 1183 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( 1  +  1 ) )  < 
( 1  +  ( 1  +  1 ) ) ) )
3837imp 418 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) )
39 readdcl 8967 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  ( 1  +  1 )  e.  RR )  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
403, 25, 39sylancl 643 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
4124, 25readdcli 8997 . . . . . . . . . 10  |-  ( 1  +  ( 1  +  1 ) )  e.  RR
4241a1i 10 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  ( 1  +  1 ) )  e.  RR )
43 lelttr 9059 . . . . . . . . 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 1183 . . . . . . . 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 451 . . . . . . 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 660 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  ( 1  +  ( 1  +  1 ) ) )
47 df-3 9952 . . . . . . 7  |-  3  =  ( 2  +  1 )
48 df-2 9951 . . . . . . . 8  |-  2  =  ( 1  +  1 )
4948oveq1i 5991 . . . . . . 7  |-  ( 2  +  1 )  =  ( ( 1  +  1 )  +  1 )
50 ax-1cn 8942 . . . . . . . 8  |-  1  e.  CC
5150, 50, 50addassi 8992 . . . . . . 7  |-  ( ( 1  +  1 )  +  1 )  =  ( 1  +  ( 1  +  1 ) )
5247, 49, 513eqtrri 2391 . . . . . 6  |-  ( 1  +  ( 1  +  1 ) )  =  3
5346, 52syl6breq 4164 . . . . 5  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 )
5453ex 423 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  <  3 ) )
5554con3d 125 . . 3  |-  ( S  e.  States  ->  ( -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3  ->  -.  ( S `  A )  <  1 ) )
56 stle1 23118 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
571, 56mpi 16 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
58 leloe 9055 . . . . . 6  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
593, 24, 58sylancl 643 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  <_ 
1  <->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) ) )
6057, 59mpbid 201 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) )
6160ord 366 . . 3  |-  ( S  e.  States  ->  ( -.  ( S `  A )  <  1  ->  ( S `  A )  =  1 ) )
6223, 55, 613syld 51 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  =  3  ->  ( S `  A )  =  1 ) )
6314, 62sylbid 206 1  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  ->  ( S `  A )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   RRcr 8883   1c1 8885    + caddc 8887    < clt 9014    <_ cle 9015   2c2 9942   3c3 9943   CHcch 21822   Statescst 21855
This theorem is referenced by:  golem2  23165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-hilex 21892
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-2 9951  df-3 9952  df-icc 10816  df-sh 22099  df-ch 22114  df-st 23104
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