Theorem rhmunitinv 13734

Description: Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)

Assertion

Ref	Expression
rhmunitinv	⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴)))

Proof of Theorem rhmunitinv

Step	Hyp	Ref	Expression
1		rhmrcl1 13711	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2		eqid 2196	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		eqid 2196	. . . . . . 7 ⊢ (invr‘𝑅) = (invr‘𝑅)
4		eqid 2196	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2196	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
6	2, 3, 4, 5	unitlinv 13682	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑅)) → (((invr‘𝑅)‘𝐴)(.r‘𝑅)𝐴) = (1r‘𝑅))
7	1, 6	sylan 283	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (((invr‘𝑅)‘𝐴)(.r‘𝑅)𝐴) = (1r‘𝑅))
8	7	fveq2d 5562	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘(((invr‘𝑅)‘𝐴)(.r‘𝑅)𝐴)) = (𝐹‘(1r‘𝑅)))
9		simpl 109	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
10		eqidd 2197	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
11		eqidd 2197	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
12	1	adantr 276	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
13		ringsrg 13603	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
14	12, 13	syl 14	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑅 ∈ SRing)
15	10, 11, 14	unitssd 13665	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑅) ⊆ (Base‘𝑅))
16	2, 3	unitinvcl 13679	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑅)) → ((invr‘𝑅)‘𝐴) ∈ (Unit‘𝑅))
17	1, 16	sylan 283	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((invr‘𝑅)‘𝐴) ∈ (Unit‘𝑅))
18	15, 17	sseldd 3184	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((invr‘𝑅)‘𝐴) ∈ (Base‘𝑅))
19		simpr 110	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Unit‘𝑅))
20	15, 19	sseldd 3184	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝐴 ∈ (Base‘𝑅))
21		eqid 2196	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2196	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
23	21, 4, 22	rhmmul 13720	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((invr‘𝑅)‘𝐴) ∈ (Base‘𝑅) ∧ 𝐴 ∈ (Base‘𝑅)) → (𝐹‘(((invr‘𝑅)‘𝐴)(.r‘𝑅)𝐴)) = ((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)))
24	9, 18, 20, 23	syl3anc 1249	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘(((invr‘𝑅)‘𝐴)(.r‘𝑅)𝐴)) = ((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)))
25		eqid 2196	. . . . . 6 ⊢ (1r‘𝑆) = (1r‘𝑆)
26	5, 25	rhm1 13723	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
27	26	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
28	8, 24, 27	3eqtr3d 2237	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (1r‘𝑆))
29		rhmrcl2 13712	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
30	29	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ Ring)
31		elrhmunit 13733	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))
32		eqid 2196	. . . . 5 ⊢ (Unit‘𝑆) = (Unit‘𝑆)
33		eqid 2196	. . . . 5 ⊢ (invr‘𝑆) = (invr‘𝑆)
34	32, 33, 22, 25	unitlinv 13682	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘𝐴) ∈ (Unit‘𝑆)) → (((invr‘𝑆)‘(𝐹‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (1r‘𝑆))
35	30, 31, 34	syl2anc 411	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (((invr‘𝑆)‘(𝐹‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (1r‘𝑆))
36	28, 35	eqtr4d 2232	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(.r‘𝑆)(𝐹‘𝐴)))
37		eqidd 2197	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((mulGrp‘𝑆) ↾s (Unit‘𝑆)) = ((mulGrp‘𝑆) ↾s (Unit‘𝑆)))
38		eqid 2196	. . . . . . . 8 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
39	38, 22	mgpplusgg 13480	. . . . . . 7 ⊢ (𝑆 ∈ Ring → (.r‘𝑆) = (+g‘(mulGrp‘𝑆)))
40	30, 39	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (.r‘𝑆) = (+g‘(mulGrp‘𝑆)))
41		basfn 12736	. . . . . . . 8 ⊢ Base Fn V
42	30	elexd 2776	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ V)
43		funfvex 5575	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
44	43	funfni 5358	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
45	41, 42, 44	sylancr 414	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑆) ∈ V)
46		eqidd 2197	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Base‘𝑆) = (Base‘𝑆))
47		eqidd 2197	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) = (Unit‘𝑆))
48		ringsrg 13603	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
49	30, 48	syl 14	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → 𝑆 ∈ SRing)
50	46, 47, 49	unitssd 13665	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) ⊆ (Base‘𝑆))
51	45, 50	ssexd 4173	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) ∈ V)
52	38	mgpex 13481	. . . . . . 7 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ V)
53	30, 52	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (mulGrp‘𝑆) ∈ V)
54	37, 40, 51, 53	ressplusgd 12806	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (.r‘𝑆) = (+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))))
55	54	oveqd 5939	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = ((𝐹‘((invr‘𝑅)‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)))
56	54	oveqd 5939	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (((invr‘𝑆)‘(𝐹‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)))
57	55, 56	eqeq12d 2211	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) ↔ ((𝐹‘((invr‘𝑅)‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴))))
58		eqid 2196	. . . . . . 7 ⊢ ((mulGrp‘𝑆) ↾s (Unit‘𝑆)) = ((mulGrp‘𝑆) ↾s (Unit‘𝑆))
59	32, 58	unitgrp 13672	. . . . . 6 ⊢ (𝑆 ∈ Ring → ((mulGrp‘𝑆) ↾s (Unit‘𝑆)) ∈ Grp)
60	29, 59	syl 14	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → ((mulGrp‘𝑆) ↾s (Unit‘𝑆)) ∈ Grp)
61	60	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((mulGrp‘𝑆) ↾s (Unit‘𝑆)) ∈ Grp)
62		elrhmunit 13733	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ((invr‘𝑅)‘𝐴) ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) ∈ (Unit‘𝑆))
63	17, 62	syldan 282	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) ∈ (Unit‘𝑆))
64	47, 37, 49	unitgrpbasd 13671	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (Unit‘𝑆) = (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))))
65	63, 64	eleqtrd 2275	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) ∈ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))))
66	32, 33	unitinvcl 13679	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘𝐴) ∈ (Unit‘𝑆)) → ((invr‘𝑆)‘(𝐹‘𝐴)) ∈ (Unit‘𝑆))
67	30, 31, 66	syl2anc 411	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((invr‘𝑆)‘(𝐹‘𝐴)) ∈ (Unit‘𝑆))
68	67, 64	eleqtrd 2275	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → ((invr‘𝑆)‘(𝐹‘𝐴)) ∈ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))))
69	31, 64	eleqtrd 2275	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))))
70		eqid 2196	. . . . 5 ⊢ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))) = (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))
71		eqid 2196	. . . . 5 ⊢ (+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))) = (+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))
72	70, 71	grprcan 13169	. . . 4 ⊢ ((((mulGrp‘𝑆) ↾s (Unit‘𝑆)) ∈ Grp ∧ ((𝐹‘((invr‘𝑅)‘𝐴)) ∈ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))) ∧ ((invr‘𝑆)‘(𝐹‘𝐴)) ∈ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))) ∧ (𝐹‘𝐴) ∈ (Base‘((mulGrp‘𝑆) ↾s (Unit‘𝑆))))) → (((𝐹‘((invr‘𝑅)‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)) ↔ (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴))))
73	61, 65, 68, 69, 72	syl13anc 1251	. . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (((𝐹‘((invr‘𝑅)‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(+g‘((mulGrp‘𝑆) ↾s (Unit‘𝑆)))(𝐹‘𝐴)) ↔ (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴))))
74	57, 73	bitrd 188	. 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (((𝐹‘((invr‘𝑅)‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) = (((invr‘𝑆)‘(𝐹‘𝐴))(.r‘𝑆)(𝐹‘𝐴)) ↔ (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴))))
75	36, 74	mpbid 147	1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  Vcvv 2763   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678   ↾_s cress 12679  +_gcplusg 12755  ._rcmulr 12756  Grpcgrp 13132  mulGrpcmgp 13476  1_rcur 13515  SRingcsrg 13519  Ringcrg 13552  Unitcui 13643  inv_rcinvr 13676   RingHom crh 13706

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-tpos 6303  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-grp 13135  df-minusg 13136  df-ghm 13371  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677  df-rhm 13708

This theorem is referenced by: (None)