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Theorem 2idlcpblrng 21164
Description: The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
Hypotheses
Ref Expression
2idlcpblrng.x 𝑋 = (Baseβ€˜π‘…)
2idlcpblrng.r 𝐸 = (𝑅 ~QG 𝑆)
2idlcpblrng.i 𝐼 = (2Idealβ€˜π‘…)
2idlcpblrng.t Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
2idlcpblrng ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ ((𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷) β†’ (𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷)))

Proof of Theorem 2idlcpblrng
StepHypRef Expression
1 simpl1 1188 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝑅 ∈ Rng)
2 simpl3 1190 . . . . . . . 8 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝑆 ∈ (SubGrpβ€˜π‘…))
3 2idlcpblrng.x . . . . . . . . 9 𝑋 = (Baseβ€˜π‘…)
4 2idlcpblrng.r . . . . . . . . 9 𝐸 = (𝑅 ~QG 𝑆)
53, 4eqger 19132 . . . . . . . 8 (𝑆 ∈ (SubGrpβ€˜π‘…) β†’ 𝐸 Er 𝑋)
62, 5syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐸 Er 𝑋)
7 simprl 769 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐴𝐸𝐢)
86, 7ersym 8730 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐢𝐸𝐴)
9 rngabl 20094 . . . . . . . 8 (𝑅 ∈ Rng β†’ 𝑅 ∈ Abel)
1093ad2ant1 1130 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ 𝑅 ∈ Abel)
11 eqid 2725 . . . . . . . . . . . 12 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
12 eqid 2725 . . . . . . . . . . . 12 (opprβ€˜π‘…) = (opprβ€˜π‘…)
13 eqid 2725 . . . . . . . . . . . 12 (LIdealβ€˜(opprβ€˜π‘…)) = (LIdealβ€˜(opprβ€˜π‘…))
14 2idlcpblrng.i . . . . . . . . . . . 12 𝐼 = (2Idealβ€˜π‘…)
1511, 12, 13, 142idlelb 21146 . . . . . . . . . . 11 (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdealβ€˜π‘…) ∧ 𝑆 ∈ (LIdealβ€˜(opprβ€˜π‘…))))
1615simplbi 496 . . . . . . . . . 10 (𝑆 ∈ 𝐼 β†’ 𝑆 ∈ (LIdealβ€˜π‘…))
17163ad2ant2 1131 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ 𝑆 ∈ (LIdealβ€˜π‘…))
1817adantr 479 . . . . . . . 8 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝑆 ∈ (LIdealβ€˜π‘…))
193, 11lidlss 21107 . . . . . . . 8 (𝑆 ∈ (LIdealβ€˜π‘…) β†’ 𝑆 βŠ† 𝑋)
2018, 19syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝑆 βŠ† 𝑋)
21 eqid 2725 . . . . . . . 8 (-gβ€˜π‘…) = (-gβ€˜π‘…)
223, 21, 4eqgabl 19788 . . . . . . 7 ((𝑅 ∈ Abel ∧ 𝑆 βŠ† 𝑋) β†’ (𝐢𝐸𝐴 ↔ (𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-gβ€˜π‘…)𝐢) ∈ 𝑆)))
2310, 20, 22syl2an2r 683 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐢𝐸𝐴 ↔ (𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-gβ€˜π‘…)𝐢) ∈ 𝑆)))
248, 23mpbid 231 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-gβ€˜π‘…)𝐢) ∈ 𝑆))
2524simp2d 1140 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐴 ∈ 𝑋)
26 simprr 771 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐡𝐸𝐷)
273, 21, 4eqgabl 19788 . . . . . . 7 ((𝑅 ∈ Abel ∧ 𝑆 βŠ† 𝑋) β†’ (𝐡𝐸𝐷 ↔ (𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-gβ€˜π‘…)𝐡) ∈ 𝑆)))
2810, 20, 27syl2an2r 683 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐡𝐸𝐷 ↔ (𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-gβ€˜π‘…)𝐡) ∈ 𝑆)))
2926, 28mpbid 231 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-gβ€˜π‘…)𝐡) ∈ 𝑆))
3029simp1d 1139 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐡 ∈ 𝑋)
31 2idlcpblrng.t . . . . 5 Β· = (.rβ€˜π‘…)
323, 31rngcl 20103 . . . 4 ((𝑅 ∈ Rng ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴 Β· 𝐡) ∈ 𝑋)
331, 25, 30, 32syl3anc 1368 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐴 Β· 𝐡) ∈ 𝑋)
3424simp1d 1139 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐢 ∈ 𝑋)
3529simp2d 1140 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝐷 ∈ 𝑋)
363, 31rngcl 20103 . . . 4 ((𝑅 ∈ Rng ∧ 𝐢 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) β†’ (𝐢 Β· 𝐷) ∈ 𝑋)
371, 34, 35, 36syl3anc 1368 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐢 Β· 𝐷) ∈ 𝑋)
38 rnggrp 20097 . . . . . . 7 (𝑅 ∈ Rng β†’ 𝑅 ∈ Grp)
39383ad2ant1 1130 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ 𝑅 ∈ Grp)
4039adantr 479 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝑅 ∈ Grp)
413, 31rngcl 20103 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝐢 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐢 Β· 𝐡) ∈ 𝑋)
421, 34, 30, 41syl3anc 1368 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐢 Β· 𝐡) ∈ 𝑋)
433, 21grpnnncan2 18992 . . . . 5 ((𝑅 ∈ Grp ∧ ((𝐢 Β· 𝐷) ∈ 𝑋 ∧ (𝐴 Β· 𝐡) ∈ 𝑋 ∧ (𝐢 Β· 𝐡) ∈ 𝑋)) β†’ (((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡))(-gβ€˜π‘…)((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡))) = ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐴 Β· 𝐡)))
4440, 37, 33, 42, 43syl13anc 1369 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡))(-gβ€˜π‘…)((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡))) = ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐴 Β· 𝐡)))
453, 31, 21, 1, 34, 35, 30rngsubdi 20110 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐢 Β· (𝐷(-gβ€˜π‘…)𝐡)) = ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡)))
46 eqid 2725 . . . . . . . . . 10 (0gβ€˜π‘…) = (0gβ€˜π‘…)
4746subg0cl 19088 . . . . . . . . 9 (𝑆 ∈ (SubGrpβ€˜π‘…) β†’ (0gβ€˜π‘…) ∈ 𝑆)
48473ad2ant3 1132 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ (0gβ€˜π‘…) ∈ 𝑆)
4948adantr 479 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (0gβ€˜π‘…) ∈ 𝑆)
5029simp3d 1141 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐷(-gβ€˜π‘…)𝐡) ∈ 𝑆)
5146, 3, 31, 11rnglidlmcl 21111 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ (LIdealβ€˜π‘…) ∧ (0gβ€˜π‘…) ∈ 𝑆) ∧ (𝐢 ∈ 𝑋 ∧ (𝐷(-gβ€˜π‘…)𝐡) ∈ 𝑆)) β†’ (𝐢 Β· (𝐷(-gβ€˜π‘…)𝐡)) ∈ 𝑆)
521, 18, 49, 34, 50, 51syl32anc 1375 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐢 Β· (𝐷(-gβ€˜π‘…)𝐡)) ∈ 𝑆)
5345, 52eqeltrrd 2826 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡)) ∈ 𝑆)
54 eqid 2725 . . . . . . . 8 (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…))
553, 31, 12, 54opprmul 20275 . . . . . . 7 (𝐡(.rβ€˜(opprβ€˜π‘…))(𝐴(-gβ€˜π‘…)𝐢)) = ((𝐴(-gβ€˜π‘…)𝐢) Β· 𝐡)
563, 31, 21, 1, 25, 34, 30rngsubdir 20111 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ ((𝐴(-gβ€˜π‘…)𝐢) Β· 𝐡) = ((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡)))
5755, 56eqtrid 2777 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐡(.rβ€˜(opprβ€˜π‘…))(𝐴(-gβ€˜π‘…)𝐢)) = ((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡)))
5812opprrng 20283 . . . . . . . . 9 (𝑅 ∈ Rng β†’ (opprβ€˜π‘…) ∈ Rng)
59583ad2ant1 1130 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ (opprβ€˜π‘…) ∈ Rng)
6059adantr 479 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (opprβ€˜π‘…) ∈ Rng)
6115simprbi 495 . . . . . . . . 9 (𝑆 ∈ 𝐼 β†’ 𝑆 ∈ (LIdealβ€˜(opprβ€˜π‘…)))
62613ad2ant2 1131 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ 𝑆 ∈ (LIdealβ€˜(opprβ€˜π‘…)))
6362adantr 479 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ 𝑆 ∈ (LIdealβ€˜(opprβ€˜π‘…)))
6424simp3d 1141 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐴(-gβ€˜π‘…)𝐢) ∈ 𝑆)
6512, 46oppr0 20287 . . . . . . . 8 (0gβ€˜π‘…) = (0gβ€˜(opprβ€˜π‘…))
6612, 3opprbas 20279 . . . . . . . 8 𝑋 = (Baseβ€˜(opprβ€˜π‘…))
6765, 66, 54, 13rnglidlmcl 21111 . . . . . . 7 ((((opprβ€˜π‘…) ∈ Rng ∧ 𝑆 ∈ (LIdealβ€˜(opprβ€˜π‘…)) ∧ (0gβ€˜π‘…) ∈ 𝑆) ∧ (𝐡 ∈ 𝑋 ∧ (𝐴(-gβ€˜π‘…)𝐢) ∈ 𝑆)) β†’ (𝐡(.rβ€˜(opprβ€˜π‘…))(𝐴(-gβ€˜π‘…)𝐢)) ∈ 𝑆)
6860, 63, 49, 30, 64, 67syl32anc 1375 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐡(.rβ€˜(opprβ€˜π‘…))(𝐴(-gβ€˜π‘…)𝐢)) ∈ 𝑆)
6957, 68eqeltrrd 2826 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ ((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡)) ∈ 𝑆)
7021subgsubcl 19091 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜π‘…) ∧ ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡)) ∈ 𝑆 ∧ ((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡)) ∈ 𝑆) β†’ (((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡))(-gβ€˜π‘…)((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡))) ∈ 𝑆)
712, 53, 69, 70syl3anc 1368 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐢 Β· 𝐡))(-gβ€˜π‘…)((𝐴 Β· 𝐡)(-gβ€˜π‘…)(𝐢 Β· 𝐡))) ∈ 𝑆)
7244, 71eqeltrrd 2826 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐴 Β· 𝐡)) ∈ 𝑆)
733, 21, 4eqgabl 19788 . . . 4 ((𝑅 ∈ Abel ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷) ↔ ((𝐴 Β· 𝐡) ∈ 𝑋 ∧ (𝐢 Β· 𝐷) ∈ 𝑋 ∧ ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐴 Β· 𝐡)) ∈ 𝑆)))
7410, 20, 73syl2an2r 683 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ ((𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷) ↔ ((𝐴 Β· 𝐡) ∈ 𝑋 ∧ (𝐢 Β· 𝐷) ∈ 𝑋 ∧ ((𝐢 Β· 𝐷)(-gβ€˜π‘…)(𝐴 Β· 𝐡)) ∈ 𝑆)))
7533, 37, 72, 74mpbir3and 1339 . 2 (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷)) β†’ (𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷))
7675ex 411 1 ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ ((𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷) β†’ (𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3941   class class class wbr 5144  β€˜cfv 6543  (class class class)co 7413   Er wer 8715  Basecbs 17174  .rcmulr 17228  0gc0g 17415  Grpcgrp 18889  -gcsg 18891  SubGrpcsubg 19074   ~QG cqg 19076  Abelcabl 19735  Rngcrng 20091  opprcoppr 20271  LIdealclidl 21101  2Idealc2idl 21142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-0g 17417  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18892  df-minusg 18893  df-sbg 18894  df-subg 19077  df-eqg 19079  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-oppr 20272  df-lss 20815  df-sra 21057  df-rgmod 21058  df-lidl 21103  df-2idl 21143
This theorem is referenced by:  2idlcpbl  21165  qus2idrng  21166  qusmulrng  21173
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