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 ahortened by AV, 9-Mar-2025.)

Ref Expression
Hypotheses 2idlcpbl.x X = Base R
2idlcpbl.r E = R ~ QG S
2idlcpbl.i I = 2Ideal R
2idlcpbl.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 2idlcpbl.x X = Base R
2 2idlcpbl.r E = R ~ QG S
3 2idlcpbl.i I = 2Ideal R
4 2idlcpbl.t · ˙ = R
5 simpll R Ring S I A E C B E D R Ring
6 eqid LIdeal R = LIdeal R
7 eqid opp r R = opp r R
8 eqid LIdeal opp r R = LIdeal opp r R
9 6 7 8 3 2idlelb S I S LIdeal R S LIdeal opp r R
10 9 simplbi S I S LIdeal R
11 10 ad2antlr R Ring S I A E C B E D S LIdeal R
12 6 lidlsubg R Ring S LIdeal R S SubGrp R
13 5 11 12 syl2anc R Ring S I A E C B E D S SubGrp R
14 1 2 eqger S SubGrp R E Er X
15 13 14 syl R Ring S I A E C B E D E Er X
16 simprl R Ring S I A E C B E D A E C
17 15 16 ersym R Ring S I A E C B E D C E A
18 ringabl R Ring R Abel
19 18 ad2antrr R Ring S I A E C B E D R Abel
20 1 3 2idlss S I S X
21 20 ad2antlr R Ring S I A E C B E D S X
22 eqid - R = - R
23 1 22 2 eqgabl R Abel S X C E A C X A X A - R C S
24 19 21 23 syl2anc R Ring S I A E C B E D C E A C X A X A - R C S
25 17 24 mpbid R Ring S I A E C B E D C X A X A - R C S
26 25 simp2d R Ring S I A E C B E D A X
27 simprr R Ring S I A E C B E D B E D
28 1 22 2 eqgabl R Abel S X B E D B X D X D - R B S
29 19 21 28 syl2anc R Ring S I A E C B E D B E D B X D X D - R B S
30 27 29 mpbid R Ring S I A E C B E D B X D X D - R B S
31 30 simp1d R Ring S I A E C B E D B X
32 1 4 5 26 31 ringcld R Ring S I A E C B E D A · ˙ B X
33 25 simp1d R Ring S I A E C B E D C X
34 30 simp2d R Ring S I A E C B E D D X
35 1 4 5 33 34 ringcld R Ring S I A E C B E D C · ˙ D X
36 ringgrp R Ring R Grp
37 36 ad2antrr R Ring S I A E C B E D R Grp
38 1 4 5 33 31 ringcld R Ring S I A E C B E D C · ˙ B X
39 1 22 grpnnncan2 R Grp C · ˙ D X A · ˙ B X C · ˙ B X C · ˙ D - R C · ˙ B - R A · ˙ B - R C · ˙ B = C · ˙ D - R A · ˙ B
40 37 35 32 38 39 syl13anc R Ring S I A E C B E D C · ˙ D - R C · ˙ B - R A · ˙ B - R C · ˙ B = C · ˙ D - R A · ˙ B
41 1 4 22 5 33 34 31 ringsubdi R Ring S I A E C B E D C · ˙ D - R B = C · ˙ D - R C · ˙ B
42 30 simp3d R Ring S I A E C B E D D - R B S
43 6 1 4 lidlmcl R Ring S LIdeal R C X D - R B S C · ˙ D - R B S
44 5 11 33 42 43 syl22anc R Ring S I A E C B E D C · ˙ D - R B S
45 41 44 eqeltrrd R Ring S I A E C B E D C · ˙ D - R C · ˙ B S
46 eqid opp r R = opp r R
47 1 4 7 46 opprmul B opp r R A - R C = A - R C · ˙ B
48 1 4 22 5 26 33 31 ringsubdir R Ring S I A E C B E D A - R C · ˙ B = A · ˙ B - R C · ˙ B
49 47 48 eqtrid R Ring S I A E C B E D B opp r R A - R C = A · ˙ B - R C · ˙ B
50 7 opprring R Ring opp r R Ring
51 50 ad2antrr R Ring S I A E C B E D opp r R Ring
52 9 simprbi S I S LIdeal opp r R
53 52 ad2antlr R Ring S I A E C B E D S LIdeal opp r R
54 25 simp3d R Ring S I A E C B E D A - R C S
55 7 1 opprbas X = Base opp r R
56 8 55 46 lidlmcl opp r R Ring S LIdeal opp r R B X A - R C S B opp r R A - R C S
57 51 53 31 54 56 syl22anc R Ring S I A E C B E D B opp r R A - R C S
58 49 57 eqeltrrd R Ring S I A E C B E D A · ˙ B - R C · ˙ B S
59 6 22 lidlsubcl R Ring S LIdeal R C · ˙ D - R C · ˙ B S A · ˙ B - R C · ˙ B S C · ˙ D - R C · ˙ B - R A · ˙ B - R C · ˙ B S
60 5 11 45 58 59 syl22anc R Ring S I A E C B E D C · ˙ D - R C · ˙ B - R A · ˙ B - R C · ˙ B S
61 40 60 eqeltrrd R Ring S I A E C B E D C · ˙ D - R A · ˙ B S
62 1 22 2 eqgabl R Abel S X A · ˙ B E C · ˙ D A · ˙ B X C · ˙ D X C · ˙ D - R A · ˙ B S
63 19 21 62 syl2anc R Ring S I A E C B E D A · ˙ B E C · ˙ D A · ˙ B X C · ˙ D X C · ˙ D - R A · ˙ B S
64 32 35 61 63 mpbir3and R Ring S I A E C B E D A · ˙ B E C · ˙ D
65 64 ex R Ring S I A E C B E D A · ˙ B E C · ˙ D