Metamath Proof Explorer


Theorem dmgmdivn0

Description: Lemma for lgamf . (Contributed by Mario Carneiro, 3-Jul-2017)

Ref Expression
Hypotheses dmgmn0.a φA
dmgmdivn0.a φM
Assertion dmgmdivn0 φAM+10

Proof

Step Hyp Ref Expression
1 dmgmn0.a φA
2 dmgmdivn0.a φM
3 1 eldifad φA
4 2 nncnd φM
5 2 nnne0d φM0
6 3 4 4 5 divdird φA+MM=AM+MM
7 4 5 dividd φMM=1
8 7 oveq2d φAM+MM=AM+1
9 6 8 eqtrd φA+MM=AM+1
10 3 4 addcld φA+M
11 2 nnnn0d φM0
12 dmgmaddn0 AM0A+M0
13 1 11 12 syl2anc φA+M0
14 10 4 13 5 divne0d φA+MM0
15 9 14 eqnetrrd φAM+10