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Theorem 2idlcpblrng 21281
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 1192 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑅 ∈ Rng)
2 simpl3 1194 . . . . . . . 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 19196 . . . . . . . 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 8757 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐶𝐸𝐴)
9 rngabl 20152 . . . . . . . 8 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
1093ad2ant1 1134 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
11 eqid 2737 . . . . . . . . . . . 12 (LIdeal‘𝑅) = (LIdeal‘𝑅)
12 eqid 2737 . . . . . . . . . . . 12 (oppr𝑅) = (oppr𝑅)
13 eqid 2737 . . . . . . . . . . . 12 (LIdeal‘(oppr𝑅)) = (LIdeal‘(oppr𝑅))
14 2idlcpblrng.i . . . . . . . . . . . 12 𝐼 = (2Ideal‘𝑅)
1511, 12, 13, 142idlelb 21263 . . . . . . . . . . 11 (𝑆𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr𝑅))))
1615simplbi 497 . . . . . . . . . 10 (𝑆𝐼𝑆 ∈ (LIdeal‘𝑅))
17163ad2ant2 1135 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘𝑅))
1817adantr 480 . . . . . . . 8 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘𝑅))
193, 11lidlss 21222 . . . . . . . 8 (𝑆 ∈ (LIdeal‘𝑅) → 𝑆𝑋)
2018, 19syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆𝑋)
21 eqid 2737 . . . . . . . 8 (-g𝑅) = (-g𝑅)
223, 21, 4eqgabl 19852 . . . . . . 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 1144 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐴𝑋)
26 simprr 773 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐵𝐸𝐷)
273, 21, 4eqgabl 19852 . . . . . . 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 1143 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐵𝑋)
31 2idlcpblrng.t . . . . 5 · = (.r𝑅)
323, 31rngcl 20161 . . . 4 ((𝑅 ∈ Rng ∧ 𝐴𝑋𝐵𝑋) → (𝐴 · 𝐵) ∈ 𝑋)
331, 25, 30, 32syl3anc 1373 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴 · 𝐵) ∈ 𝑋)
3424simp1d 1143 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐶𝑋)
3529simp2d 1144 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐷𝑋)
363, 31rngcl 20161 . . . 4 ((𝑅 ∈ Rng ∧ 𝐶𝑋𝐷𝑋) → (𝐶 · 𝐷) ∈ 𝑋)
371, 34, 35, 36syl3anc 1373 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · 𝐷) ∈ 𝑋)
38 rnggrp 20155 . . . . . . 7 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
39383ad2ant1 1134 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Grp)
4039adantr 480 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑅 ∈ Grp)
413, 31rngcl 20161 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝐶𝑋𝐵𝑋) → (𝐶 · 𝐵) ∈ 𝑋)
421, 34, 30, 41syl3anc 1373 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · 𝐵) ∈ 𝑋)
433, 21grpnnncan2 19055 . . . . 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 1374 . . . 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 20168 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g𝑅)𝐵)) = ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)))
46 eqid 2737 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
4746subg0cl 19152 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑆)
48473ad2ant3 1136 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → (0g𝑅) ∈ 𝑆)
4948adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (0g𝑅) ∈ 𝑆)
5029simp3d 1145 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐷(-g𝑅)𝐵) ∈ 𝑆)
5146, 3, 31, 11rnglidlmcl 21226 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ (LIdeal‘𝑅) ∧ (0g𝑅) ∈ 𝑆) ∧ (𝐶𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)) → (𝐶 · (𝐷(-g𝑅)𝐵)) ∈ 𝑆)
521, 18, 49, 34, 50, 51syl32anc 1380 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g𝑅)𝐵)) ∈ 𝑆)
5345, 52eqeltrrd 2842 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
54 eqid 2737 . . . . . . . 8 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
553, 31, 12, 54opprmul 20337 . . . . . . 7 (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) = ((𝐴(-g𝑅)𝐶) · 𝐵)
563, 31, 21, 1, 25, 34, 30rngsubdir 20169 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴(-g𝑅)𝐶) · 𝐵) = ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)))
5755, 56eqtrid 2789 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) = ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)))
5812opprrng 20345 . . . . . . . . 9 (𝑅 ∈ Rng → (oppr𝑅) ∈ Rng)
59583ad2ant1 1134 . . . . . . . 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 1135 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘(oppr𝑅)))
6362adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘(oppr𝑅)))
6424simp3d 1145 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴(-g𝑅)𝐶) ∈ 𝑆)
6512, 46oppr0 20349 . . . . . . . 8 (0g𝑅) = (0g‘(oppr𝑅))
6612, 3opprbas 20341 . . . . . . . 8 𝑋 = (Base‘(oppr𝑅))
6765, 66, 54, 13rnglidlmcl 21226 . . . . . . 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 1380 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) ∈ 𝑆)
6957, 68eqeltrrd 2842 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
7021subgsubcl 19155 . . . . 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 1373 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵))(-g𝑅)((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵))) ∈ 𝑆)
7244, 71eqeltrrd 2842 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)) ∈ 𝑆)
733, 21, 4eqgabl 19852 . . . 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 1343 . 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 1087   = wceq 1540  wcel 2108  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431   Er wer 8742  Basecbs 17247  .rcmulr 17298  0gc0g 17484  Grpcgrp 18951  -gcsg 18953  SubGrpcsubg 19138   ~QG cqg 19140  Abelcabl 19799  Rngcrng 20149  opprcoppr 20333  LIdealclidl 21216  2Idealc2idl 21259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-eqg 19143  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-oppr 20334  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-2idl 21260
This theorem is referenced by:  2idlcpbl  21282  qus2idrng  21283  qusmulrng  21292
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