Metamath Proof Explorer

Theorem stadd3i

Description: If the sum of 3 states is 3, then each state is 1. (Contributed by NM, 13-Nov-1999) (New usage is discouraged.)

Ref Expression
Hypotheses stle.1 
|- A e. CH
stle.2 
|- B e. CH
stm1add3.3 
|- C e. CH
Assertion stadd3i 
|- ( S e. States -> ( ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = 3 -> ( S ` A ) = 1 ) )

Proof

Step Hyp Ref Expression
1 stle.1 
 |-  A e. CH
2 stle.2 
 |-  B e. CH
3 stm1add3.3 
 |-  C e. CH
4 stcl 
 |-  ( S e. States -> ( A e. CH -> ( S ` A ) e. RR ) )
5 1 4 mpi 
 |-  ( S e. States -> ( S ` A ) e. RR )
6 5 recnd 
 |-  ( S e. States -> ( S ` A ) e. CC )
7 stcl 
 |-  ( S e. States -> ( B e. CH -> ( S ` B ) e. RR ) )
8 2 7 mpi 
 |-  ( S e. States -> ( S ` B ) e. RR )
9 8 recnd 
 |-  ( S e. States -> ( S ` B ) e. CC )
10 stcl 
 |-  ( S e. States -> ( C e. CH -> ( S ` C ) e. RR ) )
11 3 10 mpi 
 |-  ( S e. States -> ( S ` C ) e. RR )
12 11 recnd 
 |-  ( S e. States -> ( S ` C ) e. CC )
13 6 9 12 addassd 
 |-  ( S e. States -> ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) )
14 13 eqeq1d 
 |-  ( S e. States -> ( ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = 3 <-> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 ) )
15 eqcom 
 |-  ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 <-> 3 = ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) )
16 8 11 readdcld 
 |-  ( S e. States -> ( ( S ` B ) + ( S ` C ) ) e. RR )
17 5 16 readdcld 
 |-  ( S e. States -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) e. RR )
18 ltne 
 |-  ( ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) e. RR /\ ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) -> 3 =/= ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) )
19 18 ex 
 |-  ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) e. RR -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 -> 3 =/= ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) ) )
20 17 19 syl 
 |-  ( S e. States -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 -> 3 =/= ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) ) )
21 20 necon2bd 
 |-  ( S e. States -> ( 3 = ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) -> -. ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) )
22 15 21 syl5bi 
 |-  ( S e. States -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 -> -. ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) )
23 1re 
 |-  1 e. RR
24 23 23 readdcli 
 |-  ( 1 + 1 ) e. RR
25 24 a1i 
 |-  ( S e. States -> ( 1 + 1 ) e. RR )
26 1red 
 |-  ( S e. States -> 1 e. RR )
27 stle1 
 |-  ( S e. States -> ( B e. CH -> ( S ` B ) <_ 1 ) )
28 2 27 mpi 
 |-  ( S e. States -> ( S ` B ) <_ 1 )
29 stle1 
 |-  ( S e. States -> ( C e. CH -> ( S ` C ) <_ 1 ) )
30 3 29 mpi 
 |-  ( S e. States -> ( S ` C ) <_ 1 )
31 8 11 26 26 28 30 le2addd 
 |-  ( S e. States -> ( ( S ` B ) + ( S ` C ) ) <_ ( 1 + 1 ) )
32 16 25 5 31 leadd2dd 
 |-  ( S e. States -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) <_ ( ( S ` A ) + ( 1 + 1 ) ) )
33 32 adantr 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) <_ ( ( S ` A ) + ( 1 + 1 ) ) )
34 ltadd1 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( S ` A ) < 1 <-> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) ) )
35 34 biimpd 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) ) )
36 5 26 25 35 syl3anc 
 |-  ( S e. States -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) ) )
37 36 imp 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( 1 + 1 ) ) < ( 1 + ( 1 + 1 ) ) )
38 readdcl 
 |-  ( ( ( S ` A ) e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( S ` A ) + ( 1 + 1 ) ) e. RR )
39 5 24 38 sylancl 
 |-  ( S e. States -> ( ( S ` A ) + ( 1 + 1 ) ) e. RR )
40 23 24 readdcli 
 |-  ( 1 + ( 1 + 1 ) ) e. RR
41 40 a1i 
 |-  ( S e. States -> ( 1 + ( 1 + 1 ) ) e. RR )
42 lelttr 
 |-  ( ( ( ( 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 ) ) ) )
43 17 39 41 42 syl3anc 
 |-  ( 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 ) ) ) )
44 43 adantr 
 |-  ( ( 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 ) ) ) )
45 33 37 44 mp2and 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < ( 1 + ( 1 + 1 ) ) )
46 df-3 
 |-  3 = ( 2 + 1 )
47 df-2 
 |-  2 = ( 1 + 1 )
48 47 oveq1i 
 |-  ( 2 + 1 ) = ( ( 1 + 1 ) + 1 )
49 ax-1cn 
 |-  1 e. CC
50 49 49 49 addassi 
 |-  ( ( 1 + 1 ) + 1 ) = ( 1 + ( 1 + 1 ) )
51 46 48 50 3eqtrri 
 |-  ( 1 + ( 1 + 1 ) ) = 3
52 45 51 breqtrdi 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 )
53 52 ex 
 |-  ( S e. States -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 ) )
54 53 con3d 
 |-  ( S e. States -> ( -. ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) < 3 -> -. ( S ` A ) < 1 ) )
55 stle1 
 |-  ( S e. States -> ( A e. CH -> ( S ` A ) <_ 1 ) )
56 1 55 mpi 
 |-  ( S e. States -> ( S ` A ) <_ 1 )
57 leloe 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR ) -> ( ( S ` A ) <_ 1 <-> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) ) )
58 5 23 57 sylancl 
 |-  ( S e. States -> ( ( S ` A ) <_ 1 <-> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) ) )
59 56 58 mpbid 
 |-  ( S e. States -> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) )
60 59 ord 
 |-  ( S e. States -> ( -. ( S ` A ) < 1 -> ( S ` A ) = 1 ) )
61 22 54 60 3syld 
 |-  ( S e. States -> ( ( ( S ` A ) + ( ( S ` B ) + ( S ` C ) ) ) = 3 -> ( S ` A ) = 1 ) )
62 14 61 sylbid 
 |-  ( S e. States -> ( ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = 3 -> ( S ` A ) = 1 ) )