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Metamath Proof Explorer

Theorem xnn0lem1lt

Description: Extended nonnegative integer ordering relation. (Contributed by Thierry Arnoux, 30-Jul-2023)

Ref Expression
Assertion xnn0lem1lt 
|- ( ( M e. NN0 /\ N e. NN0* ) -> ( M <_ N <-> ( M - 1 ) < N ) )

Proof

Step Hyp Ref Expression
1 nn0lem1lt 
 |-  ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )
2 1 adantlr 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )
3 nn0re 
 |-  ( M e. NN0 -> M e. RR )
4 3 rexrd 
 |-  ( M e. NN0 -> M e. RR* )
5 pnfge 
 |-  ( M e. RR* -> M <_ +oo )
6 4 5 syl 
 |-  ( M e. NN0 -> M <_ +oo )
7 6 ad2antrr 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> M <_ +oo )
8 simpll 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> M e. NN0 )
9 peano2rem 
 |-  ( M e. RR -> ( M - 1 ) e. RR )
10 ltpnf 
 |-  ( ( M - 1 ) e. RR -> ( M - 1 ) < +oo )
11 8 3 9 10 4syl 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> ( M - 1 ) < +oo )
12 7 11 2thd 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> ( M <_ +oo <-> ( M - 1 ) < +oo ) )
13 xnn0nnn0pnf 
 |-  ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo )
14 13 adantll 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> N = +oo )
15 14 breq2d 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> ( M <_ N <-> M <_ +oo ) )
16 14 breq2d 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> ( ( M - 1 ) < N <-> ( M - 1 ) < +oo ) )
17 12 15 16 3bitr4d 
 |-  ( ( ( M e. NN0 /\ N e. NN0* ) /\ -. N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )
18 2 17 pm2.61dan 
 |-  ( ( M e. NN0 /\ N e. NN0* ) -> ( M <_ N <-> ( M - 1 ) < N ) )