Metamath Proof Explorer


Theorem 2idlcpbl

Description: The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015) (Proof shortened by AV, 31-Mar-2025)

Ref Expression
Hypotheses 2idlcpblrng.x X = Base R
2idlcpblrng.r E = R ~ QG S
2idlcpblrng.i I = 2Ideal R
2idlcpblrng.t · ˙ = R
Assertion 2idlcpbl R Ring S I A E C B E D A · ˙ B E C · ˙ D

Proof

Step Hyp Ref Expression
1 2idlcpblrng.x X = Base R
2 2idlcpblrng.r E = R ~ QG S
3 2idlcpblrng.i I = 2Ideal R
4 2idlcpblrng.t · ˙ = R
5 ringrng R Ring R Rng
6 5 adantr R Ring S I R Rng
7 simpr R Ring S I S I
8 eqid LIdeal R = LIdeal R
9 eqid opp r R = opp r R
10 eqid LIdeal opp r R = LIdeal opp r R
11 8 9 10 3 2idlelb S I S LIdeal R S LIdeal opp r R
12 11 simplbi S I S LIdeal R
13 8 lidlsubg R Ring S LIdeal R S SubGrp R
14 12 13 sylan2 R Ring S I S SubGrp R
15 1 2 3 4 2idlcpblrng R Rng S I S SubGrp R A E C B E D A · ˙ B E C · ˙ D
16 6 7 14 15 syl3anc R Ring S I A E C B E D A · ˙ B E C · ˙ D