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Theorem stadd3i 26871
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 26839 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
43recnd 9622 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  e.  CC )
5 stle.2 . . . . . 6  |-  B  e. 
CH
6 stcl 26839 . . . . . 6  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
75, 6mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
87recnd 9622 . . . 4  |-  ( S  e.  States  ->  ( S `  B )  e.  CC )
9 stm1add3.3 . . . . . 6  |-  C  e. 
CH
10 stcl 26839 . . . . . 6  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  e.  RR ) )
119, 10mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  C )  e.  RR )
1211recnd 9622 . . . 4  |-  ( S  e.  States  ->  ( S `  C )  e.  CC )
134, 8, 12addassd 9618 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  +  ( S `  C
) )  =  ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) ) )
1413eqeq1d 2469 . 2  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  <->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3 ) )
15 eqcom 2476 . . . 4  |-  ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3  <->  3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) )
167, 11readdcld 9623 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  e.  RR )
173, 16readdcld 9623 . . . . . 6  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  e.  RR )
18 3re 10609 . . . . . 6  |-  3  e.  RR
19 ltneOLD 9682 . . . . . . 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 1195 . . . . . 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 2682 . . . 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 9595 . . . . . . . . . . 11  |-  1  e.  RR
2524, 24readdcli 9609 . . . . . . . . . 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 26848 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
295, 28mpi 17 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
30 stle1 26848 . . . . . . . . . . 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 10170 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  <_  (
1  +  1 ) )
3316, 26, 3, 32leadd2dd 10167 . . . . . . . 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 10019 . . . . . . . . . 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 1228 . . . . . . . 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 9575 . . . . . . . . . 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 9609 . . . . . . . . . 10  |-  ( 1  +  ( 1  +  1 ) )  e.  RR
4241a1i 11 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  ( 1  +  1 ) )  e.  RR )
43 lelttr 9675 . . . . . . . . 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 1228 . . . . . . . 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 10595 . . . . . . 7  |-  3  =  ( 2  +  1 )
48 df-2 10594 . . . . . . . 8  |-  2  =  ( 1  +  1 )
4948oveq1i 6294 . . . . . . 7  |-  ( 2  +  1 )  =  ( ( 1  +  1 )  +  1 )
50 ax-1cn 9550 . . . . . . . 8  |-  1  e.  CC
5150, 50, 50addassi 9604 . . . . . . 7  |-  ( ( 1  +  1 )  +  1 )  =  ( 1  +  ( 1  +  1 ) )
5247, 49, 513eqtrri 2501 . . . . . 6  |-  ( 1  +  ( 1  +  1 ) )  =  3
5346, 52syl6breq 4486 . . . . 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 26848 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
571, 56mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
58 leloe 9671 . . . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   RRcr 9491   1c1 9493    + caddc 9495    < clt 9628    <_ cle 9629   2c2 10585   3c3 10586   CHcch 25550   Statescst 25583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-hilex 25620
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-2 10594  df-3 10595  df-icc 11536  df-sh 25828  df-ch 25843  df-st 26834
This theorem is referenced by:  golem2  26895
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