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Theorem stadd3i 29107
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 29075 . . . . . 6 |- ( S e. States -> ( A e. CH -> ( S ` A ) e. RR ) )
31, 2mpi 20 . . . . 5 |- ( S e. States -> ( S ` A ) e. RR )
43recnd 10068 . . . 4 |- ( S e. States -> ( S ` A ) e. CC )
5 stle.2 . . . . . 6 |- B e. CH
6 stcl 29075 . . . . . 6 |- ( S e. States -> ( B e. CH -> ( S ` B ) e. RR ) )
75, 6mpi 20 . . . . 5 |- ( S e. States -> ( S ` B ) e. RR )
87recnd 10068 . . . 4 |- ( S e. States -> ( S ` B ) e. CC )
9 stm1add3.3 . . . . . 6 |- C e. CH
10 stcl 29075 . . . . . 6 |- ( S e. States -> ( C e. CH -> ( S ` C ) e. RR ) )
119, 10mpi 20 . . . . 5 |- ( S e. States -> ( S ` C ) e. RR )
1211recnd 10068 . . . 4 |- ( S e. States -> ( S ` C ) e. CC )
134, 8, 12addassd 10062 . . 3 |- ( S e. States -> ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) )
1413eqeq1d 2624 . 2 |- ( S e. States -> ( ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = 3 <-> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 ) )
15 eqcom 2629 . . . 4 |- ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 <-> 3 = ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) )
167, 11readdcld 10069 . . . . . . 7 |- ( S e. States -> ( ( S ` B ) + ( S ` C ) ) e. RR )
173, 16readdcld 10069 . . . . . 6 |- ( S e. States -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) e. RR )
18 ltne 10134 . . . . . . 7 |- ( ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) e. RR /\ ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) -> 3 =/= ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) )
1918ex 450 . . . . . 6 |- ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) e. RR -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 -> 3 =/= ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) ) )
2017, 19syl 17 . . . . 5 |- ( S e. States -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 -> 3 =/= ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) ) )
2120necon2bd 2810 . . . 4 |- ( S e. States -> ( 3 = ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) -> -. ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) )
2215, 21syl5bi 232 . . 3 |- ( S e. States -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 -> -. ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) )
23 1re 10039 . . . . . . . . . . 11 |- 1 e. RR
2423, 23readdcli 10053 . . . . . . . . . 10 |- ( 1 + 1 ) e. RR
2524a1i 11 . . . . . . . . 9 |- ( S e. States -> ( 1 + 1 ) e. RR )
26 1red 10055 . . . . . . . . . 10 |- ( S e. States -> 1 e. RR )
27 stle1 29084 . . . . . . . . . . 11 |- ( S e. States -> ( B e. CH -> ( S ` B ) <_ 1 ) )
285, 27mpi 20 . . . . . . . . . 10 |- ( S e. States -> ( S ` B ) <_ 1 )
29 stle1 29084 . . . . . . . . . . 11 |- ( S e. States -> ( C e. CH -> ( S ` C ) <_ 1 ) )
309, 29mpi 20 . . . . . . . . . 10 |- ( S e. States -> ( S ` C ) <_ 1 )
317, 11, 26, 26, 28, 30le2addd 10646 . . . . . . . . 9 |- ( S e. States -> ( ( S ` B ) + ( S ` C ) ) <_ ( 1 + 1 ) )
3216, 25, 3, 31leadd2dd 10642 . . . . . . . 8 |- ( S e. States -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) <_ ( ( S ` A ) + ( 1 + 1 ) ) )
3332adantr 481 . . . . . . 7 |- ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) <_ ( ( S ` A ) + ( 1 + 1 ) ) )
34 ltadd1 10495 . . . . . . . . . 10 |- ( ( ( S ` A ) e. RR /\ 1 e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( S ` A ) < 1 <-> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) ) )
3534biimpd 219 . . . . . . . . 9 |- ( ( ( S ` A ) e. RR /\ 1 e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) ) )
363, 26, 25, 35syl3anc 1326 . . . . . . . 8 |- ( S e. States -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) ) )
3736imp 445 . . . . . . 7 |- ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) )
38 readdcl 10019 . . . . . . . . . 10 |- ( ( ( S ` A ) e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( S ` A ) + ( 1 + 1 ) ) e. RR )
393, 24, 38sylancl 694 . . . . . . . . 9 |- ( S e. States -> ( ( S ` A ) + ( 1 + 1 ) ) e. RR )
4023, 24readdcli 10053 . . . . . . . . . 10 |- ( 1 + ( 1 + 1 ) ) e. RR
4140a1i 11 . . . . . . . . 9 |- ( S e. States -> ( 1 + ( 1 + 1 ) ) e. RR )
42 lelttr 10128 . . . . . . . . 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 ) ) ) )
4317, 39, 41, 42syl3anc 1326 . . . . . . . 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 ) ) ) )
4443adantr 481 . . . . . . 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 ) ) ) )
4533, 37, 44mp2and 715 . . . . . 6 |- ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < ( 1 + ( 1 + 1 ) ) )
46 df-3 11080 . . . . . . 7 |- 3 = ( 2 + 1 )
47 df-2 11079 . . . . . . . 8 |- 2 = ( 1 + 1 )
4847oveq1i 6660 . . . . . . 7 |- ( 2 + 1 ) = ( ( 1 + 1 ) + 1 )
49 ax-1cn 9994 . . . . . . . 8 |- 1 e. CC
5049, 49, 49addassi 10048 . . . . . . 7 |- ( ( 1 + 1 ) + 1 ) = ( 1 + ( 1 + 1 ) )
5146, 48, 503eqtrri 2649 . . . . . 6 |- ( 1 + ( 1 + 1 ) ) = 3
5245, 51syl6breq 4694 . . . . 5 |- ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 )
5352ex 450 . . . 4 |- ( S e. States -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) )
5453con3d 148 . . 3 |- ( S e. States -> ( -. ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 -> -. ( S ` A ) < 1 ) )
55 stle1 29084 . . . . . 6 |- ( S e. States -> ( A e. CH -> ( S ` A ) <_ 1 ) )
561, 55mpi 20 . . . . 5 |- ( S e. States -> ( S ` A ) <_ 1 )
57 leloe 10124 . . . . . 6 |- ( ( ( S ` A ) e. RR /\ 1 e. RR ) -> ( ( S ` A ) <_ 1 <-> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) ) )
583, 23, 57sylancl 694 . . . . 5 |- ( S e. States -> ( ( S ` A ) <_ 1 <-> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) ) )
5956, 58mpbid 222 . . . 4 |- ( S e. States -> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) )
6059ord 392 . . 3 |- ( S e. States -> ( -. ( S ` A ) < 1 -> ( S ` A ) = 1 ) )
6122, 54, 603syld 60 . 2 |- ( S e. States -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 -> ( S ` A ) = 1 ) )
6214, 61sylbid 230 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 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   2c2 11070   3c3 11071   CHcch 27786   Statescst 27819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-hilex 27856
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-2 11079  df-3 11080  df-icc 12182  df-sh 28064  df-ch 28078  df-st 29070
This theorem is referenced by:  golem2  29131
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