Metamath Proof Explorer

Theorem uz11

Description: The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005)

Ref Expression
Assertion uz11 
|- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )

Proof

Step Hyp Ref Expression
1 uzid 
 |-  ( M e. ZZ -> M e. ( ZZ>= ` M ) )
2 eleq2 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) <-> M e. ( ZZ>= ` N ) ) )
3 eluzel2 
 |-  ( M e. ( ZZ>= ` N ) -> N e. ZZ )
4 2 3 syl6bi 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) -> N e. ZZ ) )
5 1 4 mpan9 
 |-  ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) -> N e. ZZ )
6 uzid 
 |-  ( N e. ZZ -> N e. ( ZZ>= ` N ) )
7 eleq2 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ( ZZ>= ` M ) <-> N e. ( ZZ>= ` N ) ) )
8 6 7 syl5ibr 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ZZ -> N e. ( ZZ>= ` M ) ) )
9 eluzle 
 |-  ( N e. ( ZZ>= ` M ) -> M <_ N )
10 8 9 syl6 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ZZ -> M <_ N ) )
11 1 2 syl5ib 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ZZ -> M e. ( ZZ>= ` N ) ) )
12 eluzle 
 |-  ( M e. ( ZZ>= ` N ) -> N <_ M )
13 11 12 syl6 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ZZ -> N <_ M ) )
14 10 13 anim12d 
 |-  ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( ( N e. ZZ /\ M e. ZZ ) -> ( M <_ N /\ N <_ M ) ) )
15 14 impl 
 |-  ( ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ N e. ZZ ) /\ M e. ZZ ) -> ( M <_ N /\ N <_ M ) )
16 15 ancoms 
 |-  ( ( M e. ZZ /\ ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ N e. ZZ ) ) -> ( M <_ N /\ N <_ M ) )
17 16 anassrs 
 |-  ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> ( M <_ N /\ N <_ M ) )
18 zre 
 |-  ( M e. ZZ -> M e. RR )
19 zre 
 |-  ( N e. ZZ -> N e. RR )
20 letri3 
 |-  ( ( M e. RR /\ N e. RR ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
21 18 19 20 syl2an 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
22 21 adantlr 
 |-  ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
23 17 22 mpbird 
 |-  ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> M = N )
24 5 23 mpdan 
 |-  ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) -> M = N )
25 24 ex 
 |-  ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> M = N ) )
26 fveq2 
 |-  ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
27 25 26 impbid1 
 |-  ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )