Step |
Hyp |
Ref |
Expression |
1 |
|
orng0le1.1 |
|- .0. = ( 0g ` F ) |
2 |
|
orng0le1.2 |
|- .1. = ( 1r ` F ) |
3 |
|
orng0le1.3 |
|- .<_ = ( le ` F ) |
4 |
|
orngring |
|- ( F e. oRing -> F e. Ring ) |
5 |
|
eqid |
|- ( Base ` F ) = ( Base ` F ) |
6 |
5 2
|
ringidcl |
|- ( F e. Ring -> .1. e. ( Base ` F ) ) |
7 |
4 6
|
syl |
|- ( F e. oRing -> .1. e. ( Base ` F ) ) |
8 |
|
eqid |
|- ( .r ` F ) = ( .r ` F ) |
9 |
5 3 1 8
|
orngsqr |
|- ( ( F e. oRing /\ .1. e. ( Base ` F ) ) -> .0. .<_ ( .1. ( .r ` F ) .1. ) ) |
10 |
7 9
|
mpdan |
|- ( F e. oRing -> .0. .<_ ( .1. ( .r ` F ) .1. ) ) |
11 |
5 8 2
|
ringlidm |
|- ( ( F e. Ring /\ .1. e. ( Base ` F ) ) -> ( .1. ( .r ` F ) .1. ) = .1. ) |
12 |
4 6 11
|
syl2anc2 |
|- ( F e. oRing -> ( .1. ( .r ` F ) .1. ) = .1. ) |
13 |
10 12
|
breqtrd |
|- ( F e. oRing -> .0. .<_ .1. ) |