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Theorem stadd3i 25587
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 25555 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  RR ) )
31, 2mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  e.  RR )
43recnd 9408 . . . 4  |-  ( S  e.  States  ->  ( S `  A )  e.  CC )
5 stle.2 . . . . . 6  |-  B  e. 
CH
6 stcl 25555 . . . . . 6  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  e.  RR ) )
75, 6mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  B )  e.  RR )
87recnd 9408 . . . 4  |-  ( S  e.  States  ->  ( S `  B )  e.  CC )
9 stm1add3.3 . . . . . 6  |-  C  e. 
CH
10 stcl 25555 . . . . . 6  |-  ( S  e.  States  ->  ( C  e. 
CH  ->  ( S `  C )  e.  RR ) )
119, 10mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  C )  e.  RR )
1211recnd 9408 . . . 4  |-  ( S  e.  States  ->  ( S `  C )  e.  CC )
134, 8, 12addassd 9404 . . 3  |-  ( S  e.  States  ->  ( ( ( S `  A )  +  ( S `  B ) )  +  ( S `  C
) )  =  ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) ) )
1413eqeq1d 2449 . 2  |-  ( S  e.  States  ->  ( ( ( ( S `  A
)  +  ( S `
 B ) )  +  ( S `  C ) )  =  3  <->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3 ) )
15 eqcom 2443 . . . 4  |-  ( ( ( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  =  3  <->  3  =  ( ( S `  A )  +  ( ( S `  B
)  +  ( S `
 C ) ) ) )
167, 11readdcld 9409 . . . . . . 7  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  e.  RR )
173, 16readdcld 9409 . . . . . 6  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  e.  RR )
18 3re 10391 . . . . . 6  |-  3  e.  RR
19 ltneOLD 9468 . . . . . . 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 1181 . . . . . 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 2658 . . . 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 9381 . . . . . . . . . . 11  |-  1  e.  RR
2524, 24readdcli 9395 . . . . . . . . . 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 25564 . . . . . . . . . . 11  |-  ( S  e.  States  ->  ( B  e. 
CH  ->  ( S `  B )  <_  1
) )
295, 28mpi 17 . . . . . . . . . 10  |-  ( S  e.  States  ->  ( S `  B )  <_  1
)
30 stle1 25564 . . . . . . . . . . 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 9953 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 B )  +  ( S `  C
) )  <_  (
1  +  1 ) )
3316, 26, 3, 32leadd2dd 9950 . . . . . . . 8  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  (
( S `  A
)  +  ( 1  +  1 ) ) )
3433adantr 462 . . . . . . 7  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <_  ( ( S `
 A )  +  ( 1  +  1 ) ) )
35 ltadd1 9802 . . . . . . . . . 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 1213 . . . . . . . 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 9361 . . . . . . . . . 10  |-  ( ( ( S `  A
)  e.  RR  /\  ( 1  +  1 )  e.  RR )  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
403, 25, 39sylancl 657 . . . . . . . . 9  |-  ( S  e.  States  ->  ( ( S `
 A )  +  ( 1  +  1 ) )  e.  RR )
4124, 25readdcli 9395 . . . . . . . . . 10  |-  ( 1  +  ( 1  +  1 ) )  e.  RR
4241a1i 11 . . . . . . . . 9  |-  ( S  e.  States  ->  ( 1  +  ( 1  +  1 ) )  e.  RR )
43 lelttr 9461 . . . . . . . . 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 1213 . . . . . . . 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 462 . . . . . . 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 674 . . . . . 6  |-  ( ( S  e.  States  /\  ( S `  A )  <  1 )  ->  (
( S `  A
)  +  ( ( S `  B )  +  ( S `  C ) ) )  <  ( 1  +  ( 1  +  1 ) ) )
47 df-3 10377 . . . . . . 7  |-  3  =  ( 2  +  1 )
48 df-2 10376 . . . . . . . 8  |-  2  =  ( 1  +  1 )
4948oveq1i 6100 . . . . . . 7  |-  ( 2  +  1 )  =  ( ( 1  +  1 )  +  1 )
50 ax-1cn 9336 . . . . . . . 8  |-  1  e.  CC
5150, 50, 50addassi 9390 . . . . . . 7  |-  ( ( 1  +  1 )  +  1 )  =  ( 1  +  ( 1  +  1 ) )
5247, 49, 513eqtrri 2466 . . . . . 6  |-  ( 1  +  ( 1  +  1 ) )  =  3
5346, 52syl6breq 4328 . . . . 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 25564 . . . . . 6  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  <_  1
) )
571, 56mpi 17 . . . . 5  |-  ( S  e.  States  ->  ( S `  A )  <_  1
)
58 leloe 9457 . . . . . 6  |-  ( ( ( S `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( S `  A )  <_  1  <->  ( ( S `  A
)  <  1  \/  ( S `  A )  =  1 ) ) )
593, 24, 58sylancl 657 . . . . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   RRcr 9277   1c1 9279    + caddc 9281    < clt 9414    <_ cle 9415   2c2 10367   3c3 10368   CHcch 24266   Statescst 24299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-hilex 24336
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-2 10376  df-3 10377  df-icc 11303  df-sh 24544  df-ch 24559  df-st 25550
This theorem is referenced by:  golem2  25611
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