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Metamath Proof Explorer

Theorem dvds1

Description: The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015)

Ref Expression
Assertion dvds1 
|- ( M e. NN0 -> ( M || 1 <-> M = 1 ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( M e. NN0 /\ M || 1 ) -> M e. NN0 )
2 1nn0 
 |-  1 e. NN0
3 2 a1i 
 |-  ( ( M e. NN0 /\ M || 1 ) -> 1 e. NN0 )
4 simpr 
 |-  ( ( M e. NN0 /\ M || 1 ) -> M || 1 )
5 nn0z 
 |-  ( M e. NN0 -> M e. ZZ )
6 1dvds 
 |-  ( M e. ZZ -> 1 || M )
7 5 6 syl 
 |-  ( M e. NN0 -> 1 || M )
8 7 adantr 
 |-  ( ( M e. NN0 /\ M || 1 ) -> 1 || M )
9 dvdseq 
 |-  ( ( ( M e. NN0 /\ 1 e. NN0 ) /\ ( M || 1 /\ 1 || M ) ) -> M = 1 )
10 1 3 4 8 9 syl22anc 
 |-  ( ( M e. NN0 /\ M || 1 ) -> M = 1 )
11 10 ex 
 |-  ( M e. NN0 -> ( M || 1 -> M = 1 ) )
12 id 
 |-  ( M = 1 -> M = 1 )
13 1z 
 |-  1 e. ZZ
14 iddvds 
 |-  ( 1 e. ZZ -> 1 || 1 )
15 13 14 ax-mp 
 |-  1 || 1
16 12 15 eqbrtrdi 
 |-  ( M = 1 -> M || 1 )
17 11 16 impbid1 
 |-  ( M e. NN0 -> ( M || 1 <-> M = 1 ) )