MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2idlcpblrng Structured version   Visualization version   GIF version

Theorem 2idlcpblrng 21159
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 1189 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑅 ∈ Rng)
2 simpl3 1191 . . . . . . . 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 19127 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝑅) → 𝐸 Er 𝑋)
62, 5syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐸 Er 𝑋)
7 simprl 770 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐴𝐸𝐶)
86, 7ersym 8731 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐶𝐸𝐴)
9 rngabl 20089 . . . . . . . 8 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
1093ad2ant1 1131 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
11 eqid 2728 . . . . . . . . . . . 12 (LIdeal‘𝑅) = (LIdeal‘𝑅)
12 eqid 2728 . . . . . . . . . . . 12 (oppr𝑅) = (oppr𝑅)
13 eqid 2728 . . . . . . . . . . . 12 (LIdeal‘(oppr𝑅)) = (LIdeal‘(oppr𝑅))
14 2idlcpblrng.i . . . . . . . . . . . 12 𝐼 = (2Ideal‘𝑅)
1511, 12, 13, 142idlelb 21141 . . . . . . . . . . 11 (𝑆𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr𝑅))))
1615simplbi 497 . . . . . . . . . 10 (𝑆𝐼𝑆 ∈ (LIdeal‘𝑅))
17163ad2ant2 1132 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘𝑅))
1817adantr 480 . . . . . . . 8 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘𝑅))
193, 11lidlss 21102 . . . . . . . 8 (𝑆 ∈ (LIdeal‘𝑅) → 𝑆𝑋)
2018, 19syl 17 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆𝑋)
21 eqid 2728 . . . . . . . 8 (-g𝑅) = (-g𝑅)
223, 21, 4eqgabl 19783 . . . . . . 7 ((𝑅 ∈ Abel ∧ 𝑆𝑋) → (𝐶𝐸𝐴 ↔ (𝐶𝑋𝐴𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆)))
2310, 20, 22syl2an2r 684 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶𝐸𝐴 ↔ (𝐶𝑋𝐴𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆)))
248, 23mpbid 231 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶𝑋𝐴𝑋 ∧ (𝐴(-g𝑅)𝐶) ∈ 𝑆))
2524simp2d 1141 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐴𝑋)
26 simprr 772 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐵𝐸𝐷)
273, 21, 4eqgabl 19783 . . . . . . 7 ((𝑅 ∈ Abel ∧ 𝑆𝑋) → (𝐵𝐸𝐷 ↔ (𝐵𝑋𝐷𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)))
2810, 20, 27syl2an2r 684 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵𝐸𝐷 ↔ (𝐵𝑋𝐷𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)))
2926, 28mpbid 231 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵𝑋𝐷𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆))
3029simp1d 1140 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐵𝑋)
31 2idlcpblrng.t . . . . 5 · = (.r𝑅)
323, 31rngcl 20098 . . . 4 ((𝑅 ∈ Rng ∧ 𝐴𝑋𝐵𝑋) → (𝐴 · 𝐵) ∈ 𝑋)
331, 25, 30, 32syl3anc 1369 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴 · 𝐵) ∈ 𝑋)
3424simp1d 1140 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐶𝑋)
3529simp2d 1141 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝐷𝑋)
363, 31rngcl 20098 . . . 4 ((𝑅 ∈ Rng ∧ 𝐶𝑋𝐷𝑋) → (𝐶 · 𝐷) ∈ 𝑋)
371, 34, 35, 36syl3anc 1369 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · 𝐷) ∈ 𝑋)
38 rnggrp 20092 . . . . . . 7 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
39383ad2ant1 1131 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Grp)
4039adantr 480 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑅 ∈ Grp)
413, 31rngcl 20098 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝐶𝑋𝐵𝑋) → (𝐶 · 𝐵) ∈ 𝑋)
421, 34, 30, 41syl3anc 1369 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · 𝐵) ∈ 𝑋)
433, 21grpnnncan2 18987 . . . . 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 1370 . . . 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 20105 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g𝑅)𝐵)) = ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)))
46 eqid 2728 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
4746subg0cl 19083 . . . . . . . . 9 (𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑆)
48473ad2ant3 1133 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → (0g𝑅) ∈ 𝑆)
4948adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (0g𝑅) ∈ 𝑆)
5029simp3d 1142 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐷(-g𝑅)𝐵) ∈ 𝑆)
5146, 3, 31, 11rnglidlmcl 21106 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆 ∈ (LIdeal‘𝑅) ∧ (0g𝑅) ∈ 𝑆) ∧ (𝐶𝑋 ∧ (𝐷(-g𝑅)𝐵) ∈ 𝑆)) → (𝐶 · (𝐷(-g𝑅)𝐵)) ∈ 𝑆)
521, 18, 49, 34, 50, 51syl32anc 1376 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g𝑅)𝐵)) ∈ 𝑆)
5345, 52eqeltrrd 2830 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
54 eqid 2728 . . . . . . . 8 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
553, 31, 12, 54opprmul 20270 . . . . . . 7 (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) = ((𝐴(-g𝑅)𝐶) · 𝐵)
563, 31, 21, 1, 25, 34, 30rngsubdir 20106 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴(-g𝑅)𝐶) · 𝐵) = ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)))
5755, 56eqtrid 2780 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) = ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)))
5812opprrng 20278 . . . . . . . . 9 (𝑅 ∈ Rng → (oppr𝑅) ∈ Rng)
59583ad2ant1 1131 . . . . . . . 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 1132 . . . . . . . 8 ((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘(oppr𝑅)))
6362adantr 480 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘(oppr𝑅)))
6424simp3d 1142 . . . . . . 7 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐴(-g𝑅)𝐶) ∈ 𝑆)
6512, 46oppr0 20282 . . . . . . . 8 (0g𝑅) = (0g‘(oppr𝑅))
6612, 3opprbas 20274 . . . . . . . 8 𝑋 = (Base‘(oppr𝑅))
6765, 66, 54, 13rnglidlmcl 21106 . . . . . . 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 1376 . . . . . 6 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (𝐵(.r‘(oppr𝑅))(𝐴(-g𝑅)𝐶)) ∈ 𝑆)
6957, 68eqeltrrd 2830 . . . . 5 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
7021subgsubcl 19086 . . . . 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 1369 . . . 4 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → (((𝐶 · 𝐷)(-g𝑅)(𝐶 · 𝐵))(-g𝑅)((𝐴 · 𝐵)(-g𝑅)(𝐶 · 𝐵))) ∈ 𝑆)
7244, 71eqeltrrd 2830 . . 3 (((𝑅 ∈ Rng ∧ 𝑆𝐼𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)) ∈ 𝑆)
733, 21, 4eqgabl 19783 . . . 4 ((𝑅 ∈ Abel ∧ 𝑆𝑋) → ((𝐴 · 𝐵)𝐸(𝐶 · 𝐷) ↔ ((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐷) ∈ 𝑋 ∧ ((𝐶 · 𝐷)(-g𝑅)(𝐴 · 𝐵)) ∈ 𝑆)))
7410, 20, 73syl2an2r 684 . . 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 1340 . 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wss 3945   class class class wbr 5143  cfv 6543  (class class class)co 7415   Er wer 8716  Basecbs 17174  .rcmulr 17228  0gc0g 17415  Grpcgrp 18884  -gcsg 18886  SubGrpcsubg 19069   ~QG cqg 19071  Abelcabl 19730  Rngcrng 20086  opprcoppr 20266  LIdealclidl 21096  2Idealc2idl 21137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  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 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  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 5144  df-opab 5206  df-mpt 5227  df-tr 5261  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-tpos 8226  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-er 8719  df-en 8959  df-dom 8960  df-sdom 8961  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 18887  df-minusg 18888  df-sbg 18889  df-subg 19072  df-eqg 19074  df-cmn 19731  df-abl 19732  df-mgp 20069  df-rng 20087  df-oppr 20267  df-lss 20810  df-sra 21052  df-rgmod 21053  df-lidl 21098  df-2idl 21138
This theorem is referenced by:  2idlcpbl  21160  qus2idrng  21161  qusmulrng  21168
  Copyright terms: Public domain W3C validator