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Theorem imadrhmcl 20678
Description: The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)
Hypotheses
Ref Expression
imadrhmcl.r 𝑅 = (𝑁s (𝐹𝑆))
imadrhmcl.0 0 = (0g𝑁)
imadrhmcl.h (𝜑𝐹 ∈ (𝑀 RingHom 𝑁))
imadrhmcl.s (𝜑𝑆 ∈ (SubDRing‘𝑀))
imadrhmcl.1 (𝜑 → ran 𝐹 ≠ { 0 })
Assertion
Ref Expression
imadrhmcl (𝜑𝑅 ∈ DivRing)

Proof of Theorem imadrhmcl
Dummy variables 𝑎 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imadrhmcl.h . . . 4 (𝜑𝐹 ∈ (𝑀 RingHom 𝑁))
2 imadrhmcl.s . . . . 5 (𝜑𝑆 ∈ (SubDRing‘𝑀))
3 sdrgsubrg 20672 . . . . 5 (𝑆 ∈ (SubDRing‘𝑀) → 𝑆 ∈ (SubRing‘𝑀))
42, 3syl 17 . . . 4 (𝜑𝑆 ∈ (SubRing‘𝑀))
5 rhmima 20536 . . . 4 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹𝑆) ∈ (SubRing‘𝑁))
61, 4, 5syl2anc 583 . . 3 (𝜑 → (𝐹𝑆) ∈ (SubRing‘𝑁))
7 imadrhmcl.r . . . 4 𝑅 = (𝑁s (𝐹𝑆))
87subrgring 20506 . . 3 ((𝐹𝑆) ∈ (SubRing‘𝑁) → 𝑅 ∈ Ring)
96, 8syl 17 . 2 (𝜑𝑅 ∈ Ring)
10 eqid 2727 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2727 . . . . . 6 (Unit‘𝑅) = (Unit‘𝑅)
1210, 11unitss 20308 . . . . 5 (Unit‘𝑅) ⊆ (Base‘𝑅)
1312a1i 11 . . . 4 (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
14 imadrhmcl.1 . . . . . 6 (𝜑 → ran 𝐹 ≠ { 0 })
15 eqid 2727 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
16 eqid 2727 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
1715, 16rhmf 20417 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
181, 17syl 17 . . . . . . . . . 10 (𝜑𝐹:(Base‘𝑀)⟶(Base‘𝑁))
1918adantr 480 . . . . . . . . 9 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
20 rhmrcl2 20409 . . . . . . . . . . . 12 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
211, 20syl 17 . . . . . . . . . . 11 (𝜑𝑁 ∈ Ring)
22 simpr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (1r𝑅) = (0g𝑅))
23 eqid 2727 . . . . . . . . . . . . . . 15 (1r𝑁) = (1r𝑁)
247, 23subrg1 20514 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ (SubRing‘𝑁) → (1r𝑁) = (1r𝑅))
256, 24syl 17 . . . . . . . . . . . . 13 (𝜑 → (1r𝑁) = (1r𝑅))
2625adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (1r𝑁) = (1r𝑅))
27 imadrhmcl.0 . . . . . . . . . . . . . . 15 0 = (0g𝑁)
287, 27subrg0 20511 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ (SubRing‘𝑁) → 0 = (0g𝑅))
296, 28syl 17 . . . . . . . . . . . . 13 (𝜑0 = (0g𝑅))
3029adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 0 = (0g𝑅))
3122, 26, 303eqtr4rd 2778 . . . . . . . . . . 11 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 0 = (1r𝑁))
3216, 27, 2301eq0ring 20460 . . . . . . . . . . 11 ((𝑁 ∈ Ring ∧ 0 = (1r𝑁)) → (Base‘𝑁) = { 0 })
3321, 31, 32syl2an2r 684 . . . . . . . . . 10 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (Base‘𝑁) = { 0 })
3433feq3d 6703 . . . . . . . . 9 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (𝐹:(Base‘𝑀)⟶(Base‘𝑁) ↔ 𝐹:(Base‘𝑀)⟶{ 0 }))
3519, 34mpbid 231 . . . . . . . 8 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 𝐹:(Base‘𝑀)⟶{ 0 })
3627fvexi 6905 . . . . . . . . 9 0 ∈ V
3736fconst2 7211 . . . . . . . 8 (𝐹:(Base‘𝑀)⟶{ 0 } ↔ 𝐹 = ((Base‘𝑀) × { 0 }))
3835, 37sylib 217 . . . . . . 7 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 𝐹 = ((Base‘𝑀) × { 0 }))
3918ffnd 6717 . . . . . . . . 9 (𝜑𝐹 Fn (Base‘𝑀))
40 sdrgrcl 20670 . . . . . . . . . . . . 13 (𝑆 ∈ (SubDRing‘𝑀) → 𝑀 ∈ DivRing)
412, 40syl 17 . . . . . . . . . . . 12 (𝜑𝑀 ∈ DivRing)
4241drngringd 20625 . . . . . . . . . . 11 (𝜑𝑀 ∈ Ring)
43 eqid 2727 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
4415, 43ring0cl 20196 . . . . . . . . . . 11 (𝑀 ∈ Ring → (0g𝑀) ∈ (Base‘𝑀))
4542, 44syl 17 . . . . . . . . . 10 (𝜑 → (0g𝑀) ∈ (Base‘𝑀))
4645ne0d 4331 . . . . . . . . 9 (𝜑 → (Base‘𝑀) ≠ ∅)
47 fconst5 7212 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑀) ∧ (Base‘𝑀) ≠ ∅) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
4839, 46, 47syl2anc 583 . . . . . . . 8 (𝜑 → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
4948adantr 480 . . . . . . 7 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
5038, 49mpbid 231 . . . . . 6 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → ran 𝐹 = { 0 })
5114, 50mteqand 3028 . . . . 5 (𝜑 → (1r𝑅) ≠ (0g𝑅))
52 eqid 2727 . . . . . . . 8 (0g𝑅) = (0g𝑅)
53 eqid 2727 . . . . . . . 8 (1r𝑅) = (1r𝑅)
5411, 52, 530unit 20328 . . . . . . 7 (𝑅 ∈ Ring → ((0g𝑅) ∈ (Unit‘𝑅) ↔ (1r𝑅) = (0g𝑅)))
559, 54syl 17 . . . . . 6 (𝜑 → ((0g𝑅) ∈ (Unit‘𝑅) ↔ (1r𝑅) = (0g𝑅)))
5655necon3bbid 2973 . . . . 5 (𝜑 → (¬ (0g𝑅) ∈ (Unit‘𝑅) ↔ (1r𝑅) ≠ (0g𝑅)))
5751, 56mpbird 257 . . . 4 (𝜑 → ¬ (0g𝑅) ∈ (Unit‘𝑅))
58 ssdifsn 4787 . . . 4 ((Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}) ↔ ((Unit‘𝑅) ⊆ (Base‘𝑅) ∧ ¬ (0g𝑅) ∈ (Unit‘𝑅)))
5913, 57, 58sylanbrc 582 . . 3 (𝜑 → (Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6039fnfund 6649 . . . . 5 (𝜑 → Fun 𝐹)
617ressbasss2 17214 . . . . . 6 (Base‘𝑅) ⊆ (𝐹𝑆)
62 eldifi 4122 . . . . . 6 (𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) → 𝑥 ∈ (Base‘𝑅))
6361, 62sselid 3976 . . . . 5 (𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) → 𝑥 ∈ (𝐹𝑆))
64 fvelima 6958 . . . . 5 ((Fun 𝐹𝑥 ∈ (𝐹𝑆)) → ∃𝑎𝑆 (𝐹𝑎) = 𝑥)
6560, 63, 64syl2an 595 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) → ∃𝑎𝑆 (𝐹𝑎) = 𝑥)
66 simprr 772 . . . . 5 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝐹𝑎) = 𝑥)
67 simprl 770 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑎𝑆)
6867fvresd 6911 . . . . . 6 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → ((𝐹𝑆)‘𝑎) = (𝐹𝑎))
69 eqid 2727 . . . . . . . . . . 11 (𝑀s 𝑆) = (𝑀s 𝑆)
7069resrhm 20533 . . . . . . . . . 10 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁))
711, 4, 70syl2anc 583 . . . . . . . . 9 (𝜑 → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁))
72 df-ima 5685 . . . . . . . . . . 11 (𝐹𝑆) = ran (𝐹𝑆)
73 eqimss2 4037 . . . . . . . . . . 11 ((𝐹𝑆) = ran (𝐹𝑆) → ran (𝐹𝑆) ⊆ (𝐹𝑆))
7472, 73mp1i 13 . . . . . . . . . 10 (𝜑 → ran (𝐹𝑆) ⊆ (𝐹𝑆))
757resrhm2b 20534 . . . . . . . . . 10 (((𝐹𝑆) ∈ (SubRing‘𝑁) ∧ ran (𝐹𝑆) ⊆ (𝐹𝑆)) → ((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁) ↔ (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅)))
766, 74, 75syl2anc 583 . . . . . . . . 9 (𝜑 → ((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁) ↔ (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅)))
7771, 76mpbid 231 . . . . . . . 8 (𝜑 → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅))
7877ad2antrr 725 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅))
79 eldifsni 4789 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) → 𝑥 ≠ (0g𝑅))
8079ad2antlr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑥 ≠ (0g𝑅))
8168adantr 480 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘𝑎) = (𝐹𝑎))
82 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → 𝑎 = (0g𝑀))
8382fveq2d 6895 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘𝑎) = ((𝐹𝑆)‘(0g𝑀)))
8469, 43subrg0 20511 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRing‘𝑀) → (0g𝑀) = (0g‘(𝑀s 𝑆)))
854, 84syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑀) = (0g‘(𝑀s 𝑆)))
8685fveq2d 6895 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑆)‘(0g𝑀)) = ((𝐹𝑆)‘(0g‘(𝑀s 𝑆))))
87 rhmghm 20416 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅) → (𝐹𝑆) ∈ ((𝑀s 𝑆) GrpHom 𝑅))
88 eqid 2727 . . . . . . . . . . . . . . 15 (0g‘(𝑀s 𝑆)) = (0g‘(𝑀s 𝑆))
8988, 52ghmid 19169 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ ((𝑀s 𝑆) GrpHom 𝑅) → ((𝐹𝑆)‘(0g‘(𝑀s 𝑆))) = (0g𝑅))
9077, 87, 893syl 18 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑆)‘(0g‘(𝑀s 𝑆))) = (0g𝑅))
9186, 90eqtrd 2767 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑆)‘(0g𝑀)) = (0g𝑅))
9291ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘(0g𝑀)) = (0g𝑅))
9383, 92eqtrd 2767 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘𝑎) = (0g𝑅))
94 simplrr 777 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → (𝐹𝑎) = 𝑥)
9581, 93, 943eqtr3rd 2776 . . . . . . . . 9 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → 𝑥 = (0g𝑅))
9680, 95mteqand 3028 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑎 ≠ (0g𝑀))
972ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑆 ∈ (SubDRing‘𝑀))
98 eqid 2727 . . . . . . . . . 10 (Unit‘(𝑀s 𝑆)) = (Unit‘(𝑀s 𝑆))
9969, 43, 98sdrgunit 20677 . . . . . . . . 9 (𝑆 ∈ (SubDRing‘𝑀) → (𝑎 ∈ (Unit‘(𝑀s 𝑆)) ↔ (𝑎𝑆𝑎 ≠ (0g𝑀))))
10097, 99syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝑎 ∈ (Unit‘(𝑀s 𝑆)) ↔ (𝑎𝑆𝑎 ≠ (0g𝑀))))
10167, 96, 100mpbir2and 712 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑎 ∈ (Unit‘(𝑀s 𝑆)))
102 elrhmunit 20442 . . . . . . 7 (((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅) ∧ 𝑎 ∈ (Unit‘(𝑀s 𝑆))) → ((𝐹𝑆)‘𝑎) ∈ (Unit‘𝑅))
10378, 101, 102syl2anc 583 . . . . . 6 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → ((𝐹𝑆)‘𝑎) ∈ (Unit‘𝑅))
10468, 103eqeltrrd 2829 . . . . 5 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝐹𝑎) ∈ (Unit‘𝑅))
10566, 104eqeltrrd 2829 . . . 4 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑥 ∈ (Unit‘𝑅))
10665, 105rexlimddv 3156 . . 3 ((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) → 𝑥 ∈ (Unit‘𝑅))
10759, 106eqelssd 3999 . 2 (𝜑 → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g𝑅)}))
10810, 11, 52isdrng 20621 . 2 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g𝑅)})))
1099, 107, 108sylanbrc 582 1 (𝜑𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wne 2935  wrex 3065  cdif 3941  wss 3944  c0 4318  {csn 4624   × cxp 5670  ran crn 5673  cres 5674  cima 5675  Fun wfun 6536   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7414  Basecbs 17173  s cress 17202  0gc0g 17414   GrpHom cghm 19160  1rcur 20114  Ringcrg 20166  Unitcui 20287   RingHom crh 20401  SubRingcsubrg 20499  DivRingcdr 20617  SubDRingcsdrg 20667
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-mulr 17240  df-0g 17416  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-mhm 18733  df-submnd 18734  df-grp 18886  df-minusg 18887  df-subg 19071  df-ghm 19161  df-cmn 19730  df-abl 19731  df-mgp 20068  df-rng 20086  df-ur 20115  df-ring 20168  df-oppr 20266  df-dvdsr 20289  df-unit 20290  df-invr 20320  df-rhm 20404  df-subrng 20476  df-subrg 20501  df-drng 20619  df-sdrg 20668
This theorem is referenced by:  rndrhmcl  32957  ricdrng1  41736
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