Theorem telgsums 18390

Description: Telescoping finitely supported group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.)

Hypotheses

Ref	Expression
telgsums.b	|- B = ( Base ` G )
telgsums.g	|- ( ph -> G e. Abel )
telgsums.m	|- .- = ( -g ` G )
telgsums.0	|- .0. = ( 0g ` G )
telgsums.f	|- ( ph -> A. k e. NN0 C e. B )
telgsums.s	|- ( ph -> S e. NN0 )
telgsums.u	|- ( ph -> A. k e. NN0 ( S < k -> C = .0. ) )

Assertion

Ref	Expression
telgsums	|- ( ph -> ( G gsumg ( i e. NN0 |-> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) ) ) = [_ 0 / k ]_ C )

Distinct variable groups:   B, i, k   C, i   i, G   S, i, k   .0. , i, k   ph, i   .- , i

Allowed substitution hints:   ph( k)   C( k)   G( k)   .- ( k)

Proof of Theorem telgsums

Step	Hyp	Ref	Expression
1		telgsums.b	. . 3 |- B = ( Base ` G )
2		telgsums.0	. . 3 |- .0. = ( 0g ` G )
3		telgsums.g	. . . 4 |- ( ph -> G e. Abel )
4		ablcmn 18199	. . . 4 |- ( G e. Abel -> G e. CMnd )
5	3, 4	syl 17	. . 3 |- ( ph -> G e. CMnd )
6		ablgrp 18198	. . . . . . 7 |- ( G e. Abel -> G e. Grp )
7	3, 6	syl 17	. . . . . 6 |- ( ph -> G e. Grp )
8	7	adantr 481	. . . . 5 |- ( ( ph /\ i e. NN0 ) -> G e. Grp )
9		simpr 477	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> i e. NN0 )
10		telgsums.f	. . . . . . 7 |- ( ph -> A. k e. NN0 C e. B )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> A. k e. NN0 C e. B )
12		rspcsbela 4006	. . . . . 6 |- ( ( i e. NN0 /\ A. k e. NN0 C e. B ) -> [_ i / k ]_ C e. B )
13	9, 11, 12	syl2anc 693	. . . . 5 |- ( ( ph /\ i e. NN0 ) -> [_ i / k ]_ C e. B )
14		peano2nn0 11333	. . . . . 6 |- ( i e. NN0 -> ( i + 1 ) e. NN0 )
15		rspcsbela 4006	. . . . . 6 |- ( ( ( i + 1 ) e. NN0 /\ A. k e. NN0 C e. B ) -> [_ ( i + 1 ) / k ]_ C e. B )
16	14, 10, 15	syl2anr 495	. . . . 5 |- ( ( ph /\ i e. NN0 ) -> [_ ( i + 1 ) / k ]_ C e. B )
17		telgsums.m	. . . . . 6 |- .- = ( -g ` G )
18	1, 17	grpsubcl 17495	. . . . 5 |- ( ( G e. Grp /\ [_ i / k ]_ C e. B /\ [_ ( i + 1 ) / k ]_ C e. B ) -> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) e. B )
19	8, 13, 16, 18	syl3anc 1326	. . . 4 |- ( ( ph /\ i e. NN0 ) -> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) e. B )
20	19	ralrimiva 2966	. . 3 |- ( ph -> A. i e. NN0 ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) e. B )
21		telgsums.s	. . 3 |- ( ph -> S e. NN0 )
22		telgsums.u	. . . . . . . . 9 |- ( ph -> A. k e. NN0 ( S < k -> C = .0. ) )
23		rspsbca 3519	. . . . . . . . . . 11 |- ( ( i e. NN0 /\ A. k e. NN0 ( S < k -> C = .0. ) ) -> [. i / k ]. ( S < k -> C = .0. ) )
24		vex 3203	. . . . . . . . . . . 12 |- i e. _V
25		sbcimg 3477	. . . . . . . . . . . . 13 |- ( i e. _V -> ( [. i / k ]. ( S < k -> C = .0. ) <-> ( [. i / k ]. S < k -> [. i / k ]. C = .0. ) ) )
26		sbcbr2g 4710	. . . . . . . . . . . . . . 15 |- ( i e. _V -> ( [. i / k ]. S < k <-> S < [_ i / k ]_ k ) )
27		csbvarg 4003	. . . . . . . . . . . . . . . 16 |- ( i e. _V -> [_ i / k ]_ k = i )
28	27	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( i e. _V -> ( S < [_ i / k ]_ k <-> S < i ) )
29	26, 28	bitrd 268	. . . . . . . . . . . . . 14 |- ( i e. _V -> ( [. i / k ]. S < k <-> S < i ) )
30		sbceq1g 3988	. . . . . . . . . . . . . 14 |- ( i e. _V -> ( [. i / k ]. C = .0. <-> [_ i / k ]_ C = .0. ) )
31	29, 30	imbi12d 334	. . . . . . . . . . . . 13 |- ( i e. _V -> ( ( [. i / k ]. S < k -> [. i / k ]. C = .0. ) <-> ( S < i -> [_ i / k ]_ C = .0. ) ) )
32	25, 31	bitrd 268	. . . . . . . . . . . 12 |- ( i e. _V -> ( [. i / k ]. ( S < k -> C = .0. ) <-> ( S < i -> [_ i / k ]_ C = .0. ) ) )
33	24, 32	ax-mp 5	. . . . . . . . . . 11 |- ( [. i / k ]. ( S < k -> C = .0. ) <-> ( S < i -> [_ i / k ]_ C = .0. ) )
34	23, 33	sylib 208	. . . . . . . . . 10 |- ( ( i e. NN0 /\ A. k e. NN0 ( S < k -> C = .0. ) ) -> ( S < i -> [_ i / k ]_ C = .0. ) )
35	34	expcom 451	. . . . . . . . 9 |- ( A. k e. NN0 ( S < k -> C = .0. ) -> ( i e. NN0 -> ( S < i -> [_ i / k ]_ C = .0. ) ) )
36	22, 35	syl 17	. . . . . . . 8 |- ( ph -> ( i e. NN0 -> ( S < i -> [_ i / k ]_ C = .0. ) ) )
37	36	imp31 448	. . . . . . 7 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> [_ i / k ]_ C = .0. )
38	21	nn0red 11352	. . . . . . . . . . . . 13 |- ( ph -> S e. RR )
39	38	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ i e. NN0 ) -> S e. RR )
40	39	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> S e. RR )
41		nn0re 11301	. . . . . . . . . . . 12 |- ( i e. NN0 -> i e. RR )
42	41	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> i e. RR )
43	14	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> ( i + 1 ) e. NN0 )
44	43	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> ( i + 1 ) e. RR )
45		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> S < i )
46	42	ltp1d 10954	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> i < ( i + 1 ) )
47	40, 42, 44, 45, 46	lttrd 10198	. . . . . . . . . 10 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> S < ( i + 1 ) )
48	47	ex 450	. . . . . . . . 9 |- ( ( ph /\ i e. NN0 ) -> ( S < i -> S < ( i + 1 ) ) )
49		rspsbca 3519	. . . . . . . . . . 11 |- ( ( ( i + 1 ) e. NN0 /\ A. k e. NN0 ( S < k -> C = .0. ) ) -> [. ( i + 1 ) / k ]. ( S < k -> C = .0. ) )
50		ovex 6678	. . . . . . . . . . . 12 |- ( i + 1 ) e. _V
51		sbcimg 3477	. . . . . . . . . . . . 13 |- ( ( i + 1 ) e. _V -> ( [. ( i + 1 ) / k ]. ( S < k -> C = .0. ) <-> ( [. ( i + 1 ) / k ]. S < k -> [. ( i + 1 ) / k ]. C = .0. ) ) )
52		sbcbr2g 4710	. . . . . . . . . . . . . . 15 |- ( ( i + 1 ) e. _V -> ( [. ( i + 1 ) / k ]. S < k <-> S < [_ ( i + 1 ) / k ]_ k ) )
53		csbvarg 4003	. . . . . . . . . . . . . . . 16 |- ( ( i + 1 ) e. _V -> [_ ( i + 1 ) / k ]_ k = ( i + 1 ) )
54	53	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( ( i + 1 ) e. _V -> ( S < [_ ( i + 1 ) / k ]_ k <-> S < ( i + 1 ) ) )
55	52, 54	bitrd 268	. . . . . . . . . . . . . 14 |- ( ( i + 1 ) e. _V -> ( [. ( i + 1 ) / k ]. S < k <-> S < ( i + 1 ) ) )
56		sbceq1g 3988	. . . . . . . . . . . . . 14 |- ( ( i + 1 ) e. _V -> ( [. ( i + 1 ) / k ]. C = .0. <-> [_ ( i + 1 ) / k ]_ C = .0. ) )
57	55, 56	imbi12d 334	. . . . . . . . . . . . 13 |- ( ( i + 1 ) e. _V -> ( ( [. ( i + 1 ) / k ]. S < k -> [. ( i + 1 ) / k ]. C = .0. ) <-> ( S < ( i + 1 ) -> [_ ( i + 1 ) / k ]_ C = .0. ) ) )
58	51, 57	bitrd 268	. . . . . . . . . . . 12 |- ( ( i + 1 ) e. _V -> ( [. ( i + 1 ) / k ]. ( S < k -> C = .0. ) <-> ( S < ( i + 1 ) -> [_ ( i + 1 ) / k ]_ C = .0. ) ) )
59	50, 58	ax-mp 5	. . . . . . . . . . 11 |- ( [. ( i + 1 ) / k ]. ( S < k -> C = .0. ) <-> ( S < ( i + 1 ) -> [_ ( i + 1 ) / k ]_ C = .0. ) )
60	49, 59	sylib 208	. . . . . . . . . 10 |- ( ( ( i + 1 ) e. NN0 /\ A. k e. NN0 ( S < k -> C = .0. ) ) -> ( S < ( i + 1 ) -> [_ ( i + 1 ) / k ]_ C = .0. ) )
61	14, 22, 60	syl2anr 495	. . . . . . . . 9 |- ( ( ph /\ i e. NN0 ) -> ( S < ( i + 1 ) -> [_ ( i + 1 ) / k ]_ C = .0. ) )
62	48, 61	syld 47	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> ( S < i -> [_ ( i + 1 ) / k ]_ C = .0. ) )
63	62	imp 445	. . . . . . 7 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> [_ ( i + 1 ) / k ]_ C = .0. )
64	37, 63	oveq12d 6668	. . . . . 6 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) = ( .0. .- .0. ) )
65	8	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> G e. Grp )
66	1, 2	grpidcl 17450	. . . . . . . 8 |- ( G e. Grp -> .0. e. B )
67	65, 66	jccir 562	. . . . . . 7 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> ( G e. Grp /\ .0. e. B ) )
68	1, 2, 17	grpsubid 17499	. . . . . . 7 |- ( ( G e. Grp /\ .0. e. B ) -> ( .0. .- .0. ) = .0. )
69	67, 68	syl 17	. . . . . 6 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> ( .0. .- .0. ) = .0. )
70	64, 69	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ i e. NN0 ) /\ S < i ) -> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) = .0. )
71	70	ex 450	. . . 4 |- ( ( ph /\ i e. NN0 ) -> ( S < i -> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) = .0. ) )
72	71	ralrimiva 2966	. . 3 |- ( ph -> A. i e. NN0 ( S < i -> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) = .0. ) )
73	1, 2, 5, 20, 21, 72	gsummptnn0fzv 18383	. 2 |- ( ph -> ( G gsumg ( i e. NN0 |-> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) ) ) = ( G gsumg ( i e. ( 0 ... S ) |-> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) ) ) )
74		fzssuz 12382	. . . . . 6 |- ( 0 ... ( S + 1 ) ) C_ ( ZZ>= ` 0 )
75	74	a1i 11	. . . . 5 |- ( ph -> ( 0 ... ( S + 1 ) ) C_ ( ZZ>= ` 0 ) )
76		nn0uz 11722	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
77	75, 76	syl6sseqr 3652	. . . 4 |- ( ph -> ( 0 ... ( S + 1 ) ) C_ NN0 )
78		ssralv 3666	. . . 4 |- ( ( 0 ... ( S + 1 ) ) C_ NN0 -> ( A. k e. NN0 C e. B -> A. k e. ( 0 ... ( S + 1 ) ) C e. B ) )
79	77, 10, 78	sylc 65	. . 3 |- ( ph -> A. k e. ( 0 ... ( S + 1 ) ) C e. B )
80	1, 3, 17, 21, 79	telgsumfz0s 18388	. 2 |- ( ph -> ( G gsumg ( i e. ( 0 ... S ) |-> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) ) ) = ( [_ 0 / k ]_ C .- [_ ( S + 1 ) / k ]_ C ) )
81		peano2nn0 11333	. . . . . 6 |- ( S e. NN0 -> ( S + 1 ) e. NN0 )
82	21, 81	syl 17	. . . . 5 |- ( ph -> ( S + 1 ) e. NN0 )
83	38	ltp1d 10954	. . . . 5 |- ( ph -> S < ( S + 1 ) )
84		rspsbca 3519	. . . . . . 7 |- ( ( ( S + 1 ) e. NN0 /\ A. k e. NN0 ( S < k -> C = .0. ) ) -> [. ( S + 1 ) / k ]. ( S < k -> C = .0. ) )
85		ovex 6678	. . . . . . . 8 |- ( S + 1 ) e. _V
86		sbcimg 3477	. . . . . . . . 9 |- ( ( S + 1 ) e. _V -> ( [. ( S + 1 ) / k ]. ( S < k -> C = .0. ) <-> ( [. ( S + 1 ) / k ]. S < k -> [. ( S + 1 ) / k ]. C = .0. ) ) )
87		sbcbr2g 4710	. . . . . . . . . . 11 |- ( ( S + 1 ) e. _V -> ( [. ( S + 1 ) / k ]. S < k <-> S < [_ ( S + 1 ) / k ]_ k ) )
88		csbvarg 4003	. . . . . . . . . . . 12 |- ( ( S + 1 ) e. _V -> [_ ( S + 1 ) / k ]_ k = ( S + 1 ) )
89	88	breq2d 4665	. . . . . . . . . . 11 |- ( ( S + 1 ) e. _V -> ( S < [_ ( S + 1 ) / k ]_ k <-> S < ( S + 1 ) ) )
90	87, 89	bitrd 268	. . . . . . . . . 10 |- ( ( S + 1 ) e. _V -> ( [. ( S + 1 ) / k ]. S < k <-> S < ( S + 1 ) ) )
91		sbceq1g 3988	. . . . . . . . . 10 |- ( ( S + 1 ) e. _V -> ( [. ( S + 1 ) / k ]. C = .0. <-> [_ ( S + 1 ) / k ]_ C = .0. ) )
92	90, 91	imbi12d 334	. . . . . . . . 9 |- ( ( S + 1 ) e. _V -> ( ( [. ( S + 1 ) / k ]. S < k -> [. ( S + 1 ) / k ]. C = .0. ) <-> ( S < ( S + 1 ) -> [_ ( S + 1 ) / k ]_ C = .0. ) ) )
93	86, 92	bitrd 268	. . . . . . . 8 |- ( ( S + 1 ) e. _V -> ( [. ( S + 1 ) / k ]. ( S < k -> C = .0. ) <-> ( S < ( S + 1 ) -> [_ ( S + 1 ) / k ]_ C = .0. ) ) )
94	85, 93	ax-mp 5	. . . . . . 7 |- ( [. ( S + 1 ) / k ]. ( S < k -> C = .0. ) <-> ( S < ( S + 1 ) -> [_ ( S + 1 ) / k ]_ C = .0. ) )
95	84, 94	sylib 208	. . . . . 6 |- ( ( ( S + 1 ) e. NN0 /\ A. k e. NN0 ( S < k -> C = .0. ) ) -> ( S < ( S + 1 ) -> [_ ( S + 1 ) / k ]_ C = .0. ) )
96	95	ex 450	. . . . 5 |- ( ( S + 1 ) e. NN0 -> ( A. k e. NN0 ( S < k -> C = .0. ) -> ( S < ( S + 1 ) -> [_ ( S + 1 ) / k ]_ C = .0. ) ) )
97	82, 22, 83, 96	syl3c 66	. . . 4 |- ( ph -> [_ ( S + 1 ) / k ]_ C = .0. )
98	97	oveq2d 6666	. . 3 |- ( ph -> ( [_ 0 / k ]_ C .- [_ ( S + 1 ) / k ]_ C ) = ( [_ 0 / k ]_ C .- .0. ) )
99		0nn0 11307	. . . . . 6 |- 0 e. NN0
100	99	a1i 11	. . . . 5 |- ( ph -> 0 e. NN0 )
101		rspcsbela 4006	. . . . 5 |- ( ( 0 e. NN0 /\ A. k e. NN0 C e. B ) -> [_ 0 / k ]_ C e. B )
102	100, 10, 101	syl2anc 693	. . . 4 |- ( ph -> [_ 0 / k ]_ C e. B )
103	1, 2, 17	grpsubid1 17500	. . . 4 |- ( ( G e. Grp /\ [_ 0 / k ]_ C e. B ) -> ( [_ 0 / k ]_ C .- .0. ) = [_ 0 / k ]_ C )
104	7, 102, 103	syl2anc 693	. . 3 |- ( ph -> ( [_ 0 / k ]_ C .- .0. ) = [_ 0 / k ]_ C )
105	98, 104	eqtrd 2656	. 2 |- ( ph -> ( [_ 0 / k ]_ C .- [_ ( S + 1 ) / k ]_ C ) = [_ 0 / k ]_ C )
106	73, 80, 105	3eqtrd 2660	1 |- ( ph -> ( G gsumg ( i e. NN0 |-> ( [_ i / k ]_ C .- [_ ( i + 1 ) / k ]_ C ) ) ) = [_ 0 / k ]_ C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   [_csb 3533   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   Grpcgrp 17422   -gcsg 17424  CMndccmn 18193   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-abl 18196

This theorem is referenced by:  telgsum  18391