Metamath Proof Explorer

Theorem modm1p1mod0

Description: If a real number modulo a positive real number equals the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals 0. (Contributed by AV, 2-Nov-2018)

Ref Expression
Assertion modm1p1mod0 
|- ( ( A e. RR /\ M e. RR+ ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 modaddmod 
 |-  ( ( A e. RR /\ 1 e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
3 1 2 mp3an2 
 |-  ( ( A e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
4 3 eqcomd 
 |-  ( ( A e. RR /\ M e. RR+ ) -> ( ( A + 1 ) mod M ) = ( ( ( A mod M ) + 1 ) mod M ) )
5 4 adantr 
 |-  ( ( ( A e. RR /\ M e. RR+ ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( A + 1 ) mod M ) = ( ( ( A mod M ) + 1 ) mod M ) )
6 oveq1 
 |-  ( ( A mod M ) = ( M - 1 ) -> ( ( A mod M ) + 1 ) = ( ( M - 1 ) + 1 ) )
7 6 oveq1d 
 |-  ( ( A mod M ) = ( M - 1 ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( ( M - 1 ) + 1 ) mod M ) )
8 rpcn 
 |-  ( M e. RR+ -> M e. CC )
9 npcan1 
 |-  ( M e. CC -> ( ( M - 1 ) + 1 ) = M )
10 8 9 syl 
 |-  ( M e. RR+ -> ( ( M - 1 ) + 1 ) = M )
11 10 oveq1d 
 |-  ( M e. RR+ -> ( ( ( M - 1 ) + 1 ) mod M ) = ( M mod M ) )
12 modid0 
 |-  ( M e. RR+ -> ( M mod M ) = 0 )
13 11 12 eqtrd 
 |-  ( M e. RR+ -> ( ( ( M - 1 ) + 1 ) mod M ) = 0 )
14 13 adantl 
 |-  ( ( A e. RR /\ M e. RR+ ) -> ( ( ( M - 1 ) + 1 ) mod M ) = 0 )
15 7 14 sylan9eqr 
 |-  ( ( ( A e. RR /\ M e. RR+ ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = 0 )
16 5 15 eqtrd 
 |-  ( ( ( A e. RR /\ M e. RR+ ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( A + 1 ) mod M ) = 0 )
17 16 ex 
 |-  ( ( A e. RR /\ M e. RR+ ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )