Metamath Proof Explorer

Theorem staddi

Description: If the sum of 2 states is 2, then each state is 1. (Contributed by NM, 12-Nov-1999) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 stle.1 
 |-  A e. CH
2 stle.2 
 |-  B e. CH
3 stcl 
 |-  ( S e. States -> ( A e. CH -> ( S ` A ) e. RR ) )
4 1 3 mpi 
 |-  ( S e. States -> ( S ` A ) e. RR )
5 stcl 
 |-  ( S e. States -> ( B e. CH -> ( S ` B ) e. RR ) )
6 2 5 mpi 
 |-  ( S e. States -> ( S ` B ) e. RR )
7 4 6 readdcld 
 |-  ( S e. States -> ( ( S ` A ) + ( S ` B ) ) e. RR )
8 ltne 
 |-  ( ( ( ( S ` A ) + ( S ` B ) ) e. RR /\ ( ( S ` A ) + ( S ` B ) ) < 2 ) -> 2 =/= ( ( S ` A ) + ( S ` B ) ) )
9 8 necomd 
 |-  ( ( ( ( S ` A ) + ( S ` B ) ) e. RR /\ ( ( S ` A ) + ( S ` B ) ) < 2 ) -> ( ( S ` A ) + ( S ` B ) ) =/= 2 )
10 7 9 sylan 
 |-  ( ( S e. States /\ ( ( S ` A ) + ( S ` B ) ) < 2 ) -> ( ( S ` A ) + ( S ` B ) ) =/= 2 )
11 10 ex 
 |-  ( S e. States -> ( ( ( S ` A ) + ( S ` B ) ) < 2 -> ( ( S ` A ) + ( S ` B ) ) =/= 2 ) )
12 11 necon2bd 
 |-  ( S e. States -> ( ( ( S ` A ) + ( S ` B ) ) = 2 -> -. ( ( S ` A ) + ( S ` B ) ) < 2 ) )
13 1re 
 |-  1 e. RR
14 13 a1i 
 |-  ( S e. States -> 1 e. RR )
15 stle1 
 |-  ( S e. States -> ( B e. CH -> ( S ` B ) <_ 1 ) )
16 2 15 mpi 
 |-  ( S e. States -> ( S ` B ) <_ 1 )
17 6 14 4 16 leadd2dd 
 |-  ( S e. States -> ( ( S ` A ) + ( S ` B ) ) <_ ( ( S ` A ) + 1 ) )
18 17 adantr 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( S ` B ) ) <_ ( ( S ` A ) + 1 ) )
19 ltadd1 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR /\ 1 e. RR ) -> ( ( S ` A ) < 1 <-> ( ( S ` A ) + 1 ) < ( 1 + 1 ) ) )
20 19 biimpd 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR /\ 1 e. RR ) -> ( ( S ` A ) < 1 -> ( ( S ` A ) + 1 ) < ( 1 + 1 ) ) )
21 4 14 14 20 syl3anc 
 |-  ( S e. States -> ( ( S ` A ) < 1 -> ( ( S ` A ) + 1 ) < ( 1 + 1 ) ) )
22 21 imp 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + 1 ) < ( 1 + 1 ) )
23 readdcl 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR ) -> ( ( S ` A ) + 1 ) e. RR )
24 4 13 23 sylancl 
 |-  ( S e. States -> ( ( S ` A ) + 1 ) e. RR )
25 13 13 readdcli 
 |-  ( 1 + 1 ) e. RR
26 25 a1i 
 |-  ( S e. States -> ( 1 + 1 ) e. RR )
27 lelttr 
 |-  ( ( ( ( S ` A ) + ( S ` B ) ) e. RR /\ ( ( S ` A ) + 1 ) e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( ( ( S ` A ) + ( S ` B ) ) <_ ( ( S ` A ) + 1 ) /\ ( ( S ` A ) + 1 ) < ( 1 + 1 ) ) -> ( ( S ` A ) + ( S ` B ) ) < ( 1 + 1 ) ) )
28 7 24 26 27 syl3anc 
 |-  ( S e. States -> ( ( ( ( S ` A ) + ( S ` B ) ) <_ ( ( S ` A ) + 1 ) /\ ( ( S ` A ) + 1 ) < ( 1 + 1 ) ) -> ( ( S ` A ) + ( S ` B ) ) < ( 1 + 1 ) ) )
29 28 adantr 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( ( ( S ` A ) + ( S ` B ) ) <_ ( ( S ` A ) + 1 ) /\ ( ( S ` A ) + 1 ) < ( 1 + 1 ) ) -> ( ( S ` A ) + ( S ` B ) ) < ( 1 + 1 ) ) )
30 18 22 29 mp2and 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( S ` B ) ) < ( 1 + 1 ) )
31 df-2 
 |-  2 = ( 1 + 1 )
32 30 31 breqtrrdi 
 |-  ( ( S e. States /\ ( S ` A ) < 1 ) -> ( ( S ` A ) + ( S ` B ) ) < 2 )
33 32 ex 
 |-  ( S e. States -> ( ( S ` A ) < 1 -> ( ( S ` A ) + ( S ` B ) ) < 2 ) )
34 33 con3d 
 |-  ( S e. States -> ( -. ( ( S ` A ) + ( S ` B ) ) < 2 -> -. ( S ` A ) < 1 ) )
35 stle1 
 |-  ( S e. States -> ( A e. CH -> ( S ` A ) <_ 1 ) )
36 1 35 mpi 
 |-  ( S e. States -> ( S ` A ) <_ 1 )
37 leloe 
 |-  ( ( ( S ` A ) e. RR /\ 1 e. RR ) -> ( ( S ` A ) <_ 1 <-> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) ) )
38 4 13 37 sylancl 
 |-  ( S e. States -> ( ( S ` A ) <_ 1 <-> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) ) )
39 36 38 mpbid 
 |-  ( S e. States -> ( ( S ` A ) < 1 \/ ( S ` A ) = 1 ) )
40 39 ord 
 |-  ( S e. States -> ( -. ( S ` A ) < 1 -> ( S ` A ) = 1 ) )
41 12 34 40 3syld 
 |-  ( S e. States -> ( ( ( S ` A ) + ( S ` B ) ) = 2 -> ( S ` A ) = 1 ) )