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Theorem stadd3i 25799
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 25767 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
43recnd 9518 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  e.  CC )
5 stle.2 . . . . . 6  |-  B  e. 
CH
6 stcl 25767 . . . . . 6  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
75, 6mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
87recnd 9518 . . . 4  |-  ( S  e.  States  ->  ( S `  B )  e.  CC )
9 stm1add3.3 . . . . . 6  |-  C  e. 
CH
10 stcl 25767 . . . . . 6  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  e.  RR ) )
119, 10mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  C )  e.  RR )
1211recnd 9518 . . . 4  |-  ( S  e.  States  ->  ( S `  C )  e.  CC )
134, 8, 12addassd 9514 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  +  ( S `  C
) )  =  ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) ) )
1413eqeq1d 2454 . 2  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  <->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3 ) )
15 eqcom 2461 . . . 4  |-  ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3  <->  3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) )
167, 11readdcld 9519 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  e.  RR )
173, 16readdcld 9519 . . . . . 6  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  e.  RR )
18 3re 10501 . . . . . 6  |-  3  e.  RR
19 ltneOLD 9578 . . . . . . 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 1187 . . . . . 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 18 . . . . 5  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  <  3  ->  3  =/=  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) ) )
2221necon2bd 2664 . . . 4  |-  ( S  e.  States  ->  ( 3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) )  ->  -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 ) )
2315, 22syl5bi 217 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  =  3  ->  -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 ) )
24 1re 9491 . . . . . . . . . . 11  |-  1  e.  RR
2524, 24readdcli 9505 . . . . . . . . . 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 25776 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
295, 28mpi 17 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
30 stle1 25776 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  <_  1
) )
319, 30mpi 17 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  C )  <_  1
)
327, 11, 27, 27, 29, 31le2addd 10063 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  <_  (
1  +  1 ) )
3316, 26, 3, 32leadd2dd 10060 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  (
( S `  A
)  +  ( 1  +  1 ) ) )
3433adantr 465 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  ( ( S `
 A )  +  ( 1  +  1 ) ) )
35 ltadd1 9912 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR  /\  (
1  +  1 )  e.  RR )  -> 
( ( S `  A )  <  1  <->  ( ( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) ) )
3635biimpd 207 . . . . . . . . 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 1219 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( 1  +  1 ) )  < 
( 1  +  ( 1  +  1 ) ) ) )
3837imp 429 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( 1  +  1 ) )  <  ( 1  +  ( 1  +  1 ) ) )
39 readdcl 9471 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  ( 1  +  1 )  e.  RR )  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
403, 25, 39sylancl 662 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
4124, 25readdcli 9505 . . . . . . . . . 10  |-  ( 1  +  ( 1  +  1 ) )  e.  RR
4241a1i 11 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  ( 1  +  1 ) )  e.  RR )
43 lelttr 9571 . . . . . . . . 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 1219 . . . . . . . 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 465 . . . . . . 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 679 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  ( 1  +  ( 1  +  1 ) ) )
47 df-3 10487 . . . . . . 7  |-  3  =  ( 2  +  1 )
48 df-2 10486 . . . . . . . 8  |-  2  =  ( 1  +  1 )
4948oveq1i 6205 . . . . . . 7  |-  ( 2  +  1 )  =  ( ( 1  +  1 )  +  1 )
50 ax-1cn 9446 . . . . . . . 8  |-  1  e.  CC
5150, 50, 50addassi 9500 . . . . . . 7  |-  ( ( 1  +  1 )  +  1 )  =  ( 1  +  ( 1  +  1 ) )
5247, 49, 513eqtrri 2486 . . . . . 6  |-  ( 1  +  ( 1  +  1 ) )  =  3
5346, 52syl6breq 4434 . . . . 5  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3 )
5453ex 434 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  ->  ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  <  3 ) )
5554con3d 133 . . 3  |-  ( S  e.  States  ->  ( -.  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  3  ->  -.  ( S `  A )  <  1 ) )
56 stle1 25776 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
571, 56mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
58 leloe 9567 . . . . . 6  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
593, 24, 58sylancl 662 . . . . 5  |-  ( S  e.  States  ->  ( ( S `
 A )  <_ 
1  <->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) ) )
6057, 59mpbid 210 . . . 4  |-  ( S  e.  States  ->  ( ( S `
 A )  <  1  \/  ( S `
 A )  =  1 ) )
6160ord 377 . . 3  |-  ( S  e.  States  ->  ( -.  ( S `  A )  <  1  ->  ( S `  A )  =  1 ) )
6223, 55, 613syld 55 . 2  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( ( S `
 B )  +  ( S `  C
) ) )  =  3  ->  ( S `  A )  =  1 ) )
6314, 62sylbid 215 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 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   RRcr 9387   1c1 9389    + caddc 9391    < clt 9524    <_ cle 9525   2c2 10477   3c3 10478   CHcch 24478   Statescst 24511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-hilex 24548
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-2 10486  df-3 10487  df-icc 11413  df-sh 24756  df-ch 24771  df-st 25762
This theorem is referenced by:  golem2  25823
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