Metamath Proof Explorer


Theorem ringlsmss2

Description: The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024)

Ref Expression
Hypotheses ringlsmss.1 B = Base R
ringlsmss.2 G = mulGrp R
ringlsmss.3 × ˙ = LSSum G
ringlsmss2.1 φ R Ring
ringlsmss2.2 φ E B
ringlsmss2.3 φ I LIdeal R
Assertion ringlsmss2 φ E × ˙ I I

Proof

Step Hyp Ref Expression
1 ringlsmss.1 B = Base R
2 ringlsmss.2 G = mulGrp R
3 ringlsmss.3 × ˙ = LSSum G
4 ringlsmss2.1 φ R Ring
5 ringlsmss2.2 φ E B
6 ringlsmss2.3 φ I LIdeal R
7 simpr φ a E × ˙ I e E i I a = e R i a = e R i
8 4 ad2antrr φ e E i I R Ring
9 6 ad2antrr φ e E i I I LIdeal R
10 5 sselda φ e E e B
11 10 adantr φ e E i I e B
12 simpr φ e E i I i I
13 eqid LIdeal R = LIdeal R
14 eqid R = R
15 13 1 14 lidlmcl R Ring I LIdeal R e B i I e R i I
16 8 9 11 12 15 syl22anc φ e E i I e R i I
17 16 adantllr φ a E × ˙ I e E i I e R i I
18 17 adantr φ a E × ˙ I e E i I a = e R i e R i I
19 7 18 eqeltrd φ a E × ˙ I e E i I a = e R i a I
20 1 13 lidlss I LIdeal R I B
21 6 20 syl φ I B
22 1 14 2 3 5 21 elringlsm φ a E × ˙ I e E i I a = e R i
23 22 biimpa φ a E × ˙ I e E i I a = e R i
24 19 23 r19.29vva φ a E × ˙ I a I
25 24 ex φ a E × ˙ I a I
26 25 ssrdv φ E × ˙ I I