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Theorem subrgunit 13298
Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
subrgugrp.1 𝑆 = (𝑅 β†Ύs 𝐴)
subrgugrp.2 π‘ˆ = (Unitβ€˜π‘…)
subrgugrp.3 𝑉 = (Unitβ€˜π‘†)
subrgunit.4 𝐼 = (invrβ€˜π‘…)
Assertion
Ref Expression
subrgunit (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)))
Proof of Theorem subrgunit
StepHypRef Expression
1 subrgugrp.1 . . . . 5 𝑆 = (𝑅 β†Ύs 𝐴)
2 subrgugrp.2 . . . . 5 π‘ˆ = (Unitβ€˜π‘…)
3 subrgugrp.3 . . . . 5 𝑉 = (Unitβ€˜π‘†)
41, 2, 3subrguss 13295 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑉 βŠ† π‘ˆ)
54sselda 3155 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ π‘ˆ)
61subrgbas 13289 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
76adantr 276 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ 𝐴 = (Baseβ€˜π‘†))
83a1i 9 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ 𝑉 = (Unitβ€˜π‘†))
91subrgring 13283 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
10 ringsrg 13155 . . . . . 6 (𝑆 ∈ Ring β†’ 𝑆 ∈ SRing)
119, 10syl 14 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ SRing)
1211adantr 276 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ 𝑆 ∈ SRing)
13 simpr 110 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ 𝑉)
147, 8, 12, 13unitcld 13208 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ 𝐴)
15 eqid 2177 . . . . . 6 (invrβ€˜π‘†) = (invrβ€˜π‘†)
16 eqid 2177 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
173, 15, 16ringinvcl 13225 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘‹) ∈ (Baseβ€˜π‘†))
189, 17sylan 283 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ ((invrβ€˜π‘†)β€˜π‘‹) ∈ (Baseβ€˜π‘†))
19 subrgunit.4 . . . . 5 𝐼 = (invrβ€˜π‘…)
201, 19, 3, 15subrginv 13296 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ (πΌβ€˜π‘‹) = ((invrβ€˜π‘†)β€˜π‘‹))
2118, 20, 73eltr4d 2261 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ (πΌβ€˜π‘‹) ∈ 𝐴)
225, 14, 213jca 1177 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑋 ∈ 𝑉) β†’ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴))
23 eqidd 2178 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
24 eqidd 2178 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (βˆ₯rβ€˜π‘†) = (βˆ₯rβ€˜π‘†))
2511adantr 276 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑆 ∈ SRing)
26 eqidd 2178 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (.rβ€˜π‘†) = (.rβ€˜π‘†))
27 simpr2 1004 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋 ∈ 𝐴)
286adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝐴 = (Baseβ€˜π‘†))
2927, 28eleqtrd 2256 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋 ∈ (Baseβ€˜π‘†))
30 simpr3 1005 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (πΌβ€˜π‘‹) ∈ 𝐴)
3130, 28eleqtrd 2256 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (πΌβ€˜π‘‹) ∈ (Baseβ€˜π‘†))
3223, 24, 25, 26, 29, 31dvdsrmuld 13196 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋(βˆ₯rβ€˜π‘†)((πΌβ€˜π‘‹)(.rβ€˜π‘†)𝑋))
33 subrgrcl 13285 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
34 simpr1 1003 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋 ∈ π‘ˆ)
35 eqid 2177 . . . . . . 7 (.rβ€˜π‘…) = (.rβ€˜π‘…)
36 eqid 2177 . . . . . . 7 (1rβ€˜π‘…) = (1rβ€˜π‘…)
372, 19, 35, 36unitlinv 13226 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ ((πΌβ€˜π‘‹)(.rβ€˜π‘…)𝑋) = (1rβ€˜π‘…))
3833, 34, 37syl2an2r 595 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ ((πΌβ€˜π‘‹)(.rβ€˜π‘…)𝑋) = (1rβ€˜π‘…))
391, 35ressmulrg 12595 . . . . . . . 8 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝑅 ∈ Ring) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
4033, 39mpdan 421 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
4140adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
4241oveqd 5889 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ ((πΌβ€˜π‘‹)(.rβ€˜π‘…)𝑋) = ((πΌβ€˜π‘‹)(.rβ€˜π‘†)𝑋))
431, 36subrg1 13290 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4443adantr 276 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (1rβ€˜π‘…) = (1rβ€˜π‘†))
4538, 42, 443eqtr3d 2218 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ ((πΌβ€˜π‘‹)(.rβ€˜π‘†)𝑋) = (1rβ€˜π‘†))
4632, 45breqtrd 4028 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋(βˆ₯rβ€˜π‘†)(1rβ€˜π‘†))
479adantr 276 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑆 ∈ Ring)
48 eqid 2177 . . . . . . 7 (opprβ€˜π‘†) = (opprβ€˜π‘†)
4948, 16opprbasg 13178 . . . . . 6 (𝑆 ∈ Ring β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘†)))
5047, 49syl 14 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (Baseβ€˜π‘†) = (Baseβ€˜(opprβ€˜π‘†)))
51 eqidd 2178 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (βˆ₯rβ€˜(opprβ€˜π‘†)) = (βˆ₯rβ€˜(opprβ€˜π‘†)))
5248opprring 13180 . . . . . 6 (𝑆 ∈ Ring β†’ (opprβ€˜π‘†) ∈ Ring)
53 ringsrg 13155 . . . . . 6 ((opprβ€˜π‘†) ∈ Ring β†’ (opprβ€˜π‘†) ∈ SRing)
5447, 52, 533syl 17 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (opprβ€˜π‘†) ∈ SRing)
55 eqidd 2178 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (.rβ€˜(opprβ€˜π‘†)) = (.rβ€˜(opprβ€˜π‘†)))
5650, 51, 54, 55, 29, 31dvdsrmuld 13196 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋(βˆ₯rβ€˜(opprβ€˜π‘†))((πΌβ€˜π‘‹)(.rβ€˜(opprβ€˜π‘†))𝑋))
57 eqid 2177 . . . . . . 7 (.rβ€˜π‘†) = (.rβ€˜π‘†)
58 eqid 2177 . . . . . . 7 (.rβ€˜(opprβ€˜π‘†)) = (.rβ€˜(opprβ€˜π‘†))
5916, 57, 48, 58opprmulg 13174 . . . . . 6 ((𝑆 ∈ Ring ∧ (πΌβ€˜π‘‹) ∈ (Baseβ€˜π‘†) ∧ 𝑋 ∈ (Baseβ€˜π‘†)) β†’ ((πΌβ€˜π‘‹)(.rβ€˜(opprβ€˜π‘†))𝑋) = (𝑋(.rβ€˜π‘†)(πΌβ€˜π‘‹)))
6047, 31, 29, 59syl3anc 1238 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ ((πΌβ€˜π‘‹)(.rβ€˜(opprβ€˜π‘†))𝑋) = (𝑋(.rβ€˜π‘†)(πΌβ€˜π‘‹)))
612, 19, 35, 36unitrinv 13227 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (1rβ€˜π‘…))
6233, 34, 61syl2an2r 595 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (1rβ€˜π‘…))
6341oveqd 5889 . . . . . 6 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (𝑋(.rβ€˜π‘…)(πΌβ€˜π‘‹)) = (𝑋(.rβ€˜π‘†)(πΌβ€˜π‘‹)))
6462, 63, 443eqtr3d 2218 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (𝑋(.rβ€˜π‘†)(πΌβ€˜π‘‹)) = (1rβ€˜π‘†))
6560, 64eqtrd 2210 . . . 4 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ ((πΌβ€˜π‘‹)(.rβ€˜(opprβ€˜π‘†))𝑋) = (1rβ€˜π‘†))
6656, 65breqtrd 4028 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘†))
673a1i 9 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑉 = (Unitβ€˜π‘†))
68 eqidd 2178 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (1rβ€˜π‘†) = (1rβ€˜π‘†))
69 eqidd 2178 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜π‘†) = (βˆ₯rβ€˜π‘†))
70 eqidd 2178 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (opprβ€˜π‘†) = (opprβ€˜π‘†))
71 eqidd 2178 . . . . 5 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (βˆ₯rβ€˜(opprβ€˜π‘†)) = (βˆ₯rβ€˜(opprβ€˜π‘†)))
7267, 68, 69, 70, 71, 11isunitd 13206 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑋 ∈ 𝑉 ↔ (𝑋(βˆ₯rβ€˜π‘†)(1rβ€˜π‘†) ∧ 𝑋(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘†))))
7372adantr 276 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ (𝑋 ∈ 𝑉 ↔ (𝑋(βˆ₯rβ€˜π‘†)(1rβ€˜π‘†) ∧ 𝑋(βˆ₯rβ€˜(opprβ€˜π‘†))(1rβ€˜π‘†))))
7446, 66, 73mpbir2and 944 . 2 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)) β†’ 𝑋 ∈ 𝑉)
7522, 74impbida 596 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐴 ∧ (πΌβ€˜π‘‹) ∈ 𝐴)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   class class class wbr 4002  β€˜cfv 5215  (class class class)co 5872  Basecbs 12454   β†Ύs cress 12455  .rcmulr 12529  1rcur 13073  SRingcsrg 13077  Ringcrg 13110  opprcoppr 13170  βˆ₯rcdsr 13186  Unitcui 13187  invrcinvr 13220  SubRingcsubrg 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-oppr 13171  df-dvdsr 13189  df-unit 13190  df-invr 13221  df-subrg 13278
This theorem is referenced by: (None)
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