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Theorem 2idlcpblrng 21299
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 = (Base‘𝑅)
2idlcpblrng.r 𝐸 = (𝑅 ~QG 𝑆)
2idlcpblrng.i 𝐼 = (2Ideal‘𝑅)
2idlcpblrng.t · = (.r𝑅)
Assertion
Ref Expression
2idlcpblrng ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

Proof of Theorem 2idlcpblrng
StepHypRef Expression
1 simpl1 1190 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑅 ∈ Rng)
2 simpl3 1192 . . . . . . . 8 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (SubGrp‘𝑅))
3 2idlcpblrng.x . . . . . . . . 9 𝑋 = (Base‘𝑅)
4 2idlcpblrng.r . . . . . . . . 9 𝐸 = (𝑅 ~QG 𝑆)
53, 4eqger 19209 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝑅) → 𝐸 Er 𝑋)
62, 5syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐸 Er 𝑋)
7 simprl 771 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐴𝐸𝐶)
86, 7ersym 8756 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐶𝐸𝐴)
9 rngabl 20173 . . . . . . . 8 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
1093ad2ant1 1132 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
11 eqid 2735 . . . . . . . . . . . 12 (LIdeal‘𝑅) = (LIdeal‘𝑅)
12 eqid 2735 . . . . . . . . . . . 12 (oppr𝑅) = (oppr𝑅)
13 eqid 2735 . . . . . . . . . . . 12 (LIdeal‘(oppr𝑅)) = (LIdeal‘(oppr𝑅))
14 2idlcpblrng.i . . . . . . . . . . . 12 𝐼 = (2Ideal‘𝑅)
1511, 12, 13, 142idlelb 21281 . . . . . . . . . . 11 (𝑆𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr𝑅))))
1615simplbi 497 . . . . . . . . . 10 (𝑆𝐼𝑆 ∈ (LIdeal‘𝑅))
17163ad2ant2 1133 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘𝑅))
1817adantr 480 . . . . . . . 8 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘𝑅))
193, 11lidlss 21240 . . . . . . . 8 (𝑆 ∈ (LIdeal‘𝑅) → 𝑆𝑋)
2018, 19syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆𝑋)
21 eqid 2735 . . . . . . . 8 (-g𝑅) = (-g𝑅)
223, 21, 4eqgabl 19867 . . . . . . 7 ((𝑅 ∈ Abel ∧ 𝑆𝑋) → (𝐶𝐸𝐴 ↔ (𝐶𝑋𝐴𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆)))
2310, 20, 22syl2an2r 685 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶𝐸𝐴 ↔ (𝐶𝑋𝐴𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆)))
248, 23mpbid 232 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶𝑋𝐴𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆))
2524simp2d 1142 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐴𝑋)
26 simprr 773 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐵𝐸𝐷)
273, 21, 4eqgabl 19867 . . . . . . 7 ((𝑅 ∈ Abel ∧ 𝑆𝑋) → (𝐵𝐸𝐷 ↔ (𝐵𝑋𝐷𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)))
2810, 20, 27syl2an2r 685 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵𝐸𝐷 ↔ (𝐵𝑋𝐷𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)))
2926, 28mpbid 232 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵𝑋𝐷𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆))
3029simp1d 1141 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐵𝑋)
31 2idlcpblrng.t . . . . 5 · = (.r𝑅)
323, 31rngcl 20182 . . . 4 ((𝑅 ∈ Rng ∧ 𝐴𝑋𝐵𝑋) → (𝐴 · 𝐵) ∈ 𝑋)
331, 25, 30, 32syl3anc 1370 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴 · 𝐵) ∈ 𝑋)
3424simp1d 1141 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐶𝑋)
3529simp2d 1142 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐷𝑋)
363, 31rngcl 20182 . . . 4 ((𝑅 ∈ Rng ∧ 𝐶𝑋𝐷𝑋) → (𝐶 · 𝐷) ∈ 𝑋)
371, 34, 35, 36syl3anc 1370 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · 𝐷) ∈ 𝑋)
38 rnggrp 20176 . . . . . . 7 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
39383ad2ant1 1132 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Grp)
4039adantr 480 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑅 ∈ Grp)
413, 31rngcl 20182 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝐶𝑋𝐵𝑋) → (𝐶 · 𝐵) ∈ 𝑋)
421, 34, 30, 41syl3anc 1370 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · 𝐵) ∈ 𝑋)
433, 21grpnnncan2 19068 . . . . 5 ((𝑅 ∈ Grp ∧ ((𝐶 · 𝐷) ∈ 𝑋 ∧ (𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐵) ∈ 𝑋)) → (((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵))(-g𝑅)((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵))) = ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)))
4440, 37, 33, 42, 43syl13anc 1371 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵))(-g𝑅)((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵))) = ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)))
453, 31, 21, 1, 34, 35, 30rngsubdi 20189 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g𝑅)𝐵)) = ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)))
46 eqid 2735 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
4746subg0cl 19165 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑆)
48473ad2ant3 1134 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → (0g𝑅) ∈ 𝑆)
4948adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (0g𝑅) ∈ 𝑆)
5029simp3d 1143 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐷(-g𝑅)𝐵) ∈ 𝑆)
5146, 3, 31, 11rnglidlmcl 21244 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ (LIdeal‘𝑅) ∧ (0g𝑅) ∈ 𝑆) ∧ (𝐶𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)) → (𝐶 · (𝐷(-g𝑅)𝐵)) ∈ 𝑆)
521, 18, 49, 34, 50, 51syl32anc 1377 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g𝑅)𝐵)) ∈ 𝑆)
5345, 52eqeltrrd 2840 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
54 eqid 2735 . . . . . . . 8 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
553, 31, 12, 54opprmul 20354 . . . . . . 7 (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) = ((𝐴(-g𝑅)𝐶) · 𝐵)
563, 31, 21, 1, 25, 34, 30rngsubdir 20190 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴(-g𝑅)𝐶) · 𝐵) = ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)))
5755, 56eqtrid 2787 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) = ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)))
5812opprrng 20362 . . . . . . . . 9 (𝑅 ∈ Rng → (oppr𝑅) ∈ Rng)
59583ad2ant1 1132 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → (oppr𝑅) ∈ Rng)
6059adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (oppr𝑅) ∈ Rng)
6115simprbi 496 . . . . . . . . 9 (𝑆𝐼𝑆 ∈ (LIdeal‘(oppr𝑅)))
62613ad2ant2 1133 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘(oppr𝑅)))
6362adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘(oppr𝑅)))
6424simp3d 1143 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴(-g𝑅)𝐶) ∈ 𝑆)
6512, 46oppr0 20366 . . . . . . . 8 (0g𝑅) = (0g‘(oppr𝑅))
6612, 3opprbas 20358 . . . . . . . 8 𝑋 = (Base‘(oppr𝑅))
6765, 66, 54, 13rnglidlmcl 21244 . . . . . . 7 ((((oppr𝑅) ∈ Rng ∧ 𝑆 ∈ (LIdeal‘(oppr𝑅)) ∧ (0g𝑅) ∈ 𝑆) ∧ (𝐵𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) ∈ 𝑆)
6860, 63, 49, 30, 64, 67syl32anc 1377 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) ∈ 𝑆)
6957, 68eqeltrrd 2840 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
7021subgsubcl 19168 . . . . 5 ((𝑆 ∈ (SubGrp‘𝑅) ∧ ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆 ∧ ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆) → (((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵))(-g𝑅)((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵))) ∈ 𝑆)
712, 53, 69, 70syl3anc 1370 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵))(-g𝑅)((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵))) ∈ 𝑆)
7244, 71eqeltrrd 2840 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)) ∈ 𝑆)
733, 21, 4eqgabl 19867 . . . 4 ((𝑅 ∈ Abel ∧ 𝑆𝑋) → ((𝐴 · 𝐵)𝐸(𝐶 · 𝐷) ↔ ((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐷) ∈ 𝑋 ∧ ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)) ∈ 𝑆)))
7410, 20, 73syl2an2r 685 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴 · 𝐵)𝐸(𝐶 · 𝐷) ↔ ((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐷) ∈ 𝑋 ∧ ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)) ∈ 𝑆)))
7533, 37, 72, 74mpbir3and 1341 . 2 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))
7675ex 412 1 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431   Er wer 8741  Basecbs 17245  .rcmulr 17299  0gc0g 17486  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151   ~QG cqg 19153  Abelcabl 19814  Rngcrng 20170  opprcoppr 20350  LIdealclidl 21234  2Idealc2idl 21277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-eqg 19156  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-oppr 20351  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-2idl 21278
This theorem is referenced by:  2idlcpbl  21300  qus2idrng  21301  qusmulrng  21310
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