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Metamath Proof Explorer

Theorem fzm1ndvds

Description: No number between 1 and M - 1 divides M . (Contributed by Mario Carneiro, 24-Jan-2015)

Ref Expression
Assertion fzm1ndvds 
|- ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M || N )

Proof

Step Hyp Ref Expression
1 elfzle2 
 |-  ( N e. ( 1 ... ( M - 1 ) ) -> N <_ ( M - 1 ) )
2 1 adantl 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> N <_ ( M - 1 ) )
3 elfzelz 
 |-  ( N e. ( 1 ... ( M - 1 ) ) -> N e. ZZ )
4 3 adantl 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> N e. ZZ )
5 nnz 
 |-  ( M e. NN -> M e. ZZ )
6 5 adantr 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> M e. ZZ )
7 zltlem1 
 |-  ( ( N e. ZZ /\ M e. ZZ ) -> ( N < M <-> N <_ ( M - 1 ) ) )
8 4 6 7 syl2anc 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> ( N < M <-> N <_ ( M - 1 ) ) )
9 2 8 mpbird 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> N < M )
10 elfznn 
 |-  ( N e. ( 1 ... ( M - 1 ) ) -> N e. NN )
11 10 adantl 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> N e. NN )
12 11 nnred 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> N e. RR )
13 nnre 
 |-  ( M e. NN -> M e. RR )
14 13 adantr 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> M e. RR )
15 12 14 ltnled 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> ( N < M <-> -. M <_ N ) )
16 9 15 mpbid 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M <_ N )
17 dvdsle 
 |-  ( ( M e. ZZ /\ N e. NN ) -> ( M || N -> M <_ N ) )
18 6 11 17 syl2anc 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> ( M || N -> M <_ N ) )
19 16 18 mtod 
 |-  ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M || N )