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Theorem imadrhmcl 20798
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 20792 . . . . 5 (𝑆 ∈ (SubDRing‘𝑀) → 𝑆 ∈ (SubRing‘𝑀))
42, 3syl 17 . . . 4 (𝜑𝑆 ∈ (SubRing‘𝑀))
5 rhmima 20604 . . . 4 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹𝑆) ∈ (SubRing‘𝑁))
61, 4, 5syl2anc 584 . . 3 (𝜑 → (𝐹𝑆) ∈ (SubRing‘𝑁))
7 imadrhmcl.r . . . 4 𝑅 = (𝑁s (𝐹𝑆))
87subrgring 20574 . . 3 ((𝐹𝑆) ∈ (SubRing‘𝑁) → 𝑅 ∈ Ring)
96, 8syl 17 . 2 (𝜑𝑅 ∈ Ring)
10 eqid 2737 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2737 . . . . . 6 (Unit‘𝑅) = (Unit‘𝑅)
1210, 11unitss 20376 . . . . 5 (Unit‘𝑅) ⊆ (Base‘𝑅)
1312a1i 11 . . . 4 (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
14 imadrhmcl.1 . . . . . 6 (𝜑 → ran 𝐹 ≠ { 0 })
15 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
16 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
1715, 16rhmf 20485 . . . . . . . . . . 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 20477 . . . . . . . . . . . 12 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
211, 20syl 17 . . . . . . . . . . 11 (𝜑𝑁 ∈ Ring)
22 simpr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (1r𝑅) = (0g𝑅))
23 eqid 2737 . . . . . . . . . . . . . . 15 (1r𝑁) = (1r𝑁)
247, 23subrg1 20582 . . . . . . . . . . . . . 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 20579 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ (SubRing‘𝑁) → 0 = (0g𝑅))
296, 28syl 17 . . . . . . . . . . . . 13 (𝜑0 = (0g𝑅))
3029adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 0 = (0g𝑅))
3122, 26, 303eqtr4rd 2788 . . . . . . . . . . 11 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 0 = (1r𝑁))
3216, 27, 2301eq0ring 20530 . . . . . . . . . . 11 ((𝑁 ∈ Ring ∧ 0 = (1r𝑁)) → (Base‘𝑁) = { 0 })
3321, 31, 32syl2an2r 685 . . . . . . . . . 10 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (Base‘𝑁) = { 0 })
3433feq3d 6723 . . . . . . . . 9 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (𝐹:(Base‘𝑀)⟶(Base‘𝑁) ↔ 𝐹:(Base‘𝑀)⟶{ 0 }))
3519, 34mpbid 232 . . . . . . . 8 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 𝐹:(Base‘𝑀)⟶{ 0 })
3627fvexi 6920 . . . . . . . . 9 0 ∈ V
3736fconst2 7225 . . . . . . . 8 (𝐹:(Base‘𝑀)⟶{ 0 } ↔ 𝐹 = ((Base‘𝑀) × { 0 }))
3835, 37sylib 218 . . . . . . 7 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → 𝐹 = ((Base‘𝑀) × { 0 }))
3918ffnd 6737 . . . . . . . . 9 (𝜑𝐹 Fn (Base‘𝑀))
40 sdrgrcl 20790 . . . . . . . . . . . . 13 (𝑆 ∈ (SubDRing‘𝑀) → 𝑀 ∈ DivRing)
412, 40syl 17 . . . . . . . . . . . 12 (𝜑𝑀 ∈ DivRing)
4241drngringd 20737 . . . . . . . . . . 11 (𝜑𝑀 ∈ Ring)
43 eqid 2737 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
4415, 43ring0cl 20264 . . . . . . . . . . 11 (𝑀 ∈ Ring → (0g𝑀) ∈ (Base‘𝑀))
4542, 44syl 17 . . . . . . . . . 10 (𝜑 → (0g𝑀) ∈ (Base‘𝑀))
4645ne0d 4342 . . . . . . . . 9 (𝜑 → (Base‘𝑀) ≠ ∅)
47 fconst5 7226 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑀) ∧ (Base‘𝑀) ≠ ∅) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
4839, 46, 47syl2anc 584 . . . . . . . 8 (𝜑 → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
4948adantr 480 . . . . . . 7 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → (𝐹 = ((Base‘𝑀) × { 0 }) ↔ ran 𝐹 = { 0 }))
5038, 49mpbid 232 . . . . . 6 ((𝜑 ∧ (1r𝑅) = (0g𝑅)) → ran 𝐹 = { 0 })
5114, 50mteqand 3033 . . . . 5 (𝜑 → (1r𝑅) ≠ (0g𝑅))
52 eqid 2737 . . . . . . . 8 (0g𝑅) = (0g𝑅)
53 eqid 2737 . . . . . . . 8 (1r𝑅) = (1r𝑅)
5411, 52, 530unit 20396 . . . . . . 7 (𝑅 ∈ Ring → ((0g𝑅) ∈ (Unit‘𝑅) ↔ (1r𝑅) = (0g𝑅)))
559, 54syl 17 . . . . . 6 (𝜑 → ((0g𝑅) ∈ (Unit‘𝑅) ↔ (1r𝑅) = (0g𝑅)))
5655necon3bbid 2978 . . . . 5 (𝜑 → (¬ (0g𝑅) ∈ (Unit‘𝑅) ↔ (1r𝑅) ≠ (0g𝑅)))
5751, 56mpbird 257 . . . 4 (𝜑 → ¬ (0g𝑅) ∈ (Unit‘𝑅))
58 ssdifsn 4788 . . . 4 ((Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}) ↔ ((Unit‘𝑅) ⊆ (Base‘𝑅) ∧ ¬ (0g𝑅) ∈ (Unit‘𝑅)))
5913, 57, 58sylanbrc 583 . . 3 (𝜑 → (Unit‘𝑅) ⊆ ((Base‘𝑅) ∖ {(0g𝑅)}))
6039fnfund 6669 . . . . 5 (𝜑 → Fun 𝐹)
617ressbasss2 17286 . . . . . 6 (Base‘𝑅) ⊆ (𝐹𝑆)
62 eldifi 4131 . . . . . 6 (𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) → 𝑥 ∈ (Base‘𝑅))
6361, 62sselid 3981 . . . . 5 (𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) → 𝑥 ∈ (𝐹𝑆))
64 fvelima 6974 . . . . 5 ((Fun 𝐹𝑥 ∈ (𝐹𝑆)) → ∃𝑎𝑆 (𝐹𝑎) = 𝑥)
6560, 63, 64syl2an 596 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) → ∃𝑎𝑆 (𝐹𝑎) = 𝑥)
66 simprr 773 . . . . 5 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝐹𝑎) = 𝑥)
67 simprl 771 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑎𝑆)
6867fvresd 6926 . . . . . 6 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → ((𝐹𝑆)‘𝑎) = (𝐹𝑎))
69 eqid 2737 . . . . . . . . . . 11 (𝑀s 𝑆) = (𝑀s 𝑆)
7069resrhm 20601 . . . . . . . . . 10 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁))
711, 4, 70syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁))
72 df-ima 5698 . . . . . . . . . . 11 (𝐹𝑆) = ran (𝐹𝑆)
73 eqimss2 4043 . . . . . . . . . . 11 ((𝐹𝑆) = ran (𝐹𝑆) → ran (𝐹𝑆) ⊆ (𝐹𝑆))
7472, 73mp1i 13 . . . . . . . . . 10 (𝜑 → ran (𝐹𝑆) ⊆ (𝐹𝑆))
757resrhm2b 20602 . . . . . . . . . 10 (((𝐹𝑆) ∈ (SubRing‘𝑁) ∧ ran (𝐹𝑆) ⊆ (𝐹𝑆)) → ((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁) ↔ (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅)))
766, 74, 75syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑁) ↔ (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅)))
7771, 76mpbid 232 . . . . . . . 8 (𝜑 → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅))
7877ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅))
79 eldifsni 4790 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) → 𝑥 ≠ (0g𝑅))
8079ad2antlr 727 . . . . . . . . 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 6910 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘𝑎) = ((𝐹𝑆)‘(0g𝑀)))
8469, 43subrg0 20579 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRing‘𝑀) → (0g𝑀) = (0g‘(𝑀s 𝑆)))
854, 84syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑀) = (0g‘(𝑀s 𝑆)))
8685fveq2d 6910 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑆)‘(0g𝑀)) = ((𝐹𝑆)‘(0g‘(𝑀s 𝑆))))
87 rhmghm 20484 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅) → (𝐹𝑆) ∈ ((𝑀s 𝑆) GrpHom 𝑅))
88 eqid 2737 . . . . . . . . . . . . . . 15 (0g‘(𝑀s 𝑆)) = (0g‘(𝑀s 𝑆))
8988, 52ghmid 19240 . . . . . . . . . . . . . 14 ((𝐹𝑆) ∈ ((𝑀s 𝑆) GrpHom 𝑅) → ((𝐹𝑆)‘(0g‘(𝑀s 𝑆))) = (0g𝑅))
9077, 87, 893syl 18 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑆)‘(0g‘(𝑀s 𝑆))) = (0g𝑅))
9186, 90eqtrd 2777 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝑆)‘(0g𝑀)) = (0g𝑅))
9291ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘(0g𝑀)) = (0g𝑅))
9383, 92eqtrd 2777 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → ((𝐹𝑆)‘𝑎) = (0g𝑅))
94 simplrr 778 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → (𝐹𝑎) = 𝑥)
9581, 93, 943eqtr3rd 2786 . . . . . . . . 9 ((((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) ∧ 𝑎 = (0g𝑀)) → 𝑥 = (0g𝑅))
9680, 95mteqand 3033 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑎 ≠ (0g𝑀))
972ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑆 ∈ (SubDRing‘𝑀))
98 eqid 2737 . . . . . . . . . 10 (Unit‘(𝑀s 𝑆)) = (Unit‘(𝑀s 𝑆))
9969, 43, 98sdrgunit 20797 . . . . . . . . 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 713 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑎 ∈ (Unit‘(𝑀s 𝑆)))
102 elrhmunit 20510 . . . . . . 7 (((𝐹𝑆) ∈ ((𝑀s 𝑆) RingHom 𝑅) ∧ 𝑎 ∈ (Unit‘(𝑀s 𝑆))) → ((𝐹𝑆)‘𝑎) ∈ (Unit‘𝑅))
10378, 101, 102syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → ((𝐹𝑆)‘𝑎) ∈ (Unit‘𝑅))
10468, 103eqeltrrd 2842 . . . . 5 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → (𝐹𝑎) ∈ (Unit‘𝑅))
10566, 104eqeltrrd 2842 . . . 4 (((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) ∧ (𝑎𝑆 ∧ (𝐹𝑎) = 𝑥)) → 𝑥 ∈ (Unit‘𝑅))
10665, 105rexlimddv 3161 . . 3 ((𝜑𝑥 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) → 𝑥 ∈ (Unit‘𝑅))
10759, 106eqelssd 4005 . 2 (𝜑 → (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g𝑅)}))
10810, 11, 52isdrng 20733 . 2 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g𝑅)})))
1099, 107, 108sylanbrc 583 1 (𝜑𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cdif 3948  wss 3951  c0 4333  {csn 4626   × cxp 5683  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  0gc0g 17484   GrpHom cghm 19230  1rcur 20178  Ringcrg 20230  Unitcui 20355   RingHom crh 20469  SubRingcsubrg 20569  DivRingcdr 20729  SubDRingcsdrg 20787
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-map 8868  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-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-drng 20731  df-sdrg 20788
This theorem is referenced by:  rndrhmcl  33299  ricdrng1  42538
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