Theorem nn0sumshdiglem1 42415

Description: Lemma 1 for nn0sumshdig 42417 (induction step). (Contributed by AV, 7-Jun-2020.)

Assertion

Ref	Expression
nn0sumshdiglem1	|- ( y e. NN -> ( A. a e. NN0 ( (#b ` a ) = y -> a = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) -> A. a e. NN0 ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )

Distinct variable group:   k, a, y

Proof of Theorem nn0sumshdiglem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( a = x -> (#b ` a ) = (#b ` x ) )
2	1	eqeq1d 2624	. . . 4 |- ( a = x -> ( (#b ` a ) = y <-> (#b ` x ) = y ) )
3		id 22	. . . . 5 |- ( a = x -> a = x )
4		oveq2 6658	. . . . . . 7 |- ( a = x -> ( k (digit ` 2 ) a ) = ( k (digit ` 2 ) x ) )
5	4	oveq1d 6665	. . . . . 6 |- ( a = x -> ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) = ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) )
6	5	sumeq2sdv 14435	. . . . 5 |- ( a = x -> sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) )
7	3, 6	eqeq12d 2637	. . . 4 |- ( a = x -> ( a = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) <-> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) )
8	2, 7	imbi12d 334	. . 3 |- ( a = x -> ( ( (#b ` a ) = y -> a = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) <-> ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) )
9	8	cbvralv 3171	. 2 |- ( A. a e. NN0 ( (#b ` a ) = y -> a = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) <-> A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) )
10		elnn0 11294	. . . . . 6 |- ( a e. NN0 <-> ( a e. NN \/ a = 0 ) )
11		nn0sumshdiglemA 42413	. . . . . . . . 9 |- ( ( ( a e. NN /\ ( a / 2 ) e. NN ) /\ y e. NN ) -> ( A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
12	11	expimpd 629	. . . . . . . 8 |- ( ( a e. NN /\ ( a / 2 ) e. NN ) -> ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
13		nn0sumshdiglemB 42414	. . . . . . . . 9 |- ( ( ( a e. NN /\ ( ( a - 1 ) / 2 ) e. NN0 ) /\ y e. NN ) -> ( A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
14	13	expimpd 629	. . . . . . . 8 |- ( ( a e. NN /\ ( ( a - 1 ) / 2 ) e. NN0 ) -> ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
15		nneom 42321	. . . . . . . 8 |- ( a e. NN -> ( ( a / 2 ) e. NN \/ ( ( a - 1 ) / 2 ) e. NN0 ) )
16	12, 14, 15	mpjaodan 827	. . . . . . 7 |- ( a e. NN -> ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
17		eqcom 2629	. . . . . . . . . . . . . 14 |- ( 1 = ( y + 1 ) <-> ( y + 1 ) = 1 )
18	17	a1i 11	. . . . . . . . . . . . 13 |- ( y e. NN -> ( 1 = ( y + 1 ) <-> ( y + 1 ) = 1 ) )
19		nncn 11028	. . . . . . . . . . . . . 14 |- ( y e. NN -> y e. CC )
20		1cnd 10056	. . . . . . . . . . . . . 14 |- ( y e. NN -> 1 e. CC )
21	19, 20, 20	addlsub 10447	. . . . . . . . . . . . 13 |- ( y e. NN -> ( ( y + 1 ) = 1 <-> y = ( 1 - 1 ) ) )
22		1m1e0 11089	. . . . . . . . . . . . . . 15 |- ( 1 - 1 ) = 0
23	22	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. NN -> ( 1 - 1 ) = 0 )
24	23	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( y e. NN -> ( y = ( 1 - 1 ) <-> y = 0 ) )
25	18, 21, 24	3bitrd 294	. . . . . . . . . . . 12 |- ( y e. NN -> ( 1 = ( y + 1 ) <-> y = 0 ) )
26		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( y = 0 -> ( y + 1 ) = ( 0 + 1 ) )
27	26	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( y = 0 -> ( 0..^ ( y + 1 ) ) = ( 0..^ ( 0 + 1 ) ) )
28		0p1e1 11132	. . . . . . . . . . . . . . . . 17 |- ( 0 + 1 ) = 1
29	28	oveq2i 6661	. . . . . . . . . . . . . . . 16 |- ( 0..^ ( 0 + 1 ) ) = ( 0..^ 1 )
30		fzo01 12550	. . . . . . . . . . . . . . . 16 |- ( 0..^ 1 ) = { 0 }
31	29, 30	eqtri 2644	. . . . . . . . . . . . . . 15 |- ( 0..^ ( 0 + 1 ) ) = { 0 }
32	27, 31	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( y = 0 -> ( 0..^ ( y + 1 ) ) = { 0 } )
33	32	sumeq1d 14431	. . . . . . . . . . . . 13 |- ( y = 0 -> sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) = sum_ k e. { 0 } ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) )
34		0cn 10032	. . . . . . . . . . . . . 14 |- 0 e. CC
35		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k (digit ` 2 ) 0 ) = ( 0 (digit ` 2 ) 0 ) )
36		2nn 11185	. . . . . . . . . . . . . . . . . . 19 |- 2 e. NN
37		0z 11388	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
38		dig0 42400	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. NN /\ 0 e. ZZ ) -> ( 0 (digit ` 2 ) 0 ) = 0 )
39	36, 37, 38	mp2an 708	. . . . . . . . . . . . . . . . . 18 |- ( 0 (digit ` 2 ) 0 ) = 0
40	35, 39	syl6eq 2672	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( k (digit ` 2 ) 0 ) = 0 )
41		oveq2 6658	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( 2 ^ k ) = ( 2 ^ 0 ) )
42		2cn 11091	. . . . . . . . . . . . . . . . . . 19 |- 2 e. CC
43		exp0 12864	. . . . . . . . . . . . . . . . . . 19 |- ( 2 e. CC -> ( 2 ^ 0 ) = 1 )
44	42, 43	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( 2 ^ 0 ) = 1
45	41, 44	syl6eq 2672	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( 2 ^ k ) = 1 )
46	40, 45	oveq12d 6668	. . . . . . . . . . . . . . . 16 |- ( k = 0 -> ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) = ( 0 x. 1 ) )
47		1re 10039	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
48		mul02lem2 10213	. . . . . . . . . . . . . . . . 17 |- ( 1 e. RR -> ( 0 x. 1 ) = 0 )
49	47, 48	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( 0 x. 1 ) = 0
50	46, 49	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) = 0 )
51	50	sumsn 14475	. . . . . . . . . . . . . 14 |- ( ( 0 e. CC /\ 0 e. CC ) -> sum_ k e. { 0 } ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) = 0 )
52	34, 34, 51	mp2an 708	. . . . . . . . . . . . 13 |- sum_ k e. { 0 } ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) = 0
53	33, 52	syl6req 2673	. . . . . . . . . . . 12 |- ( y = 0 -> 0 = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) )
54	25, 53	syl6bi 243	. . . . . . . . . . 11 |- ( y e. NN -> ( 1 = ( y + 1 ) -> 0 = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) ) )
55	54	adantl 482	. . . . . . . . . 10 |- ( ( a = 0 /\ y e. NN ) -> ( 1 = ( y + 1 ) -> 0 = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) ) )
56		fveq2 6191	. . . . . . . . . . . . . 14 |- ( a = 0 -> (#b ` a ) = (#b ` 0 ) )
57		blen0 42366	. . . . . . . . . . . . . 14 |- (#b ` 0 ) = 1
58	56, 57	syl6eq 2672	. . . . . . . . . . . . 13 |- ( a = 0 -> (#b ` a ) = 1 )
59	58	eqeq1d 2624	. . . . . . . . . . . 12 |- ( a = 0 -> ( (#b ` a ) = ( y + 1 ) <-> 1 = ( y + 1 ) ) )
60		id 22	. . . . . . . . . . . . 13 |- ( a = 0 -> a = 0 )
61		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( a = 0 -> ( k (digit ` 2 ) a ) = ( k (digit ` 2 ) 0 ) )
62	61	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( a = 0 -> ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) = ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) )
63	62	sumeq2sdv 14435	. . . . . . . . . . . . 13 |- ( a = 0 -> sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) )
64	60, 63	eqeq12d 2637	. . . . . . . . . . . 12 |- ( a = 0 -> ( a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) <-> 0 = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) ) )
65	59, 64	imbi12d 334	. . . . . . . . . . 11 |- ( a = 0 -> ( ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) <-> ( 1 = ( y + 1 ) -> 0 = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) ) ) )
66	65	adantr 481	. . . . . . . . . 10 |- ( ( a = 0 /\ y e. NN ) -> ( ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) <-> ( 1 = ( y + 1 ) -> 0 = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) 0 ) x. ( 2 ^ k ) ) ) ) )
67	55, 66	mpbird 247	. . . . . . . . 9 |- ( ( a = 0 /\ y e. NN ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) )
68	67	a1d 25	. . . . . . . 8 |- ( ( a = 0 /\ y e. NN ) -> ( A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
69	68	expimpd 629	. . . . . . 7 |- ( a = 0 -> ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
70	16, 69	jaoi 394	. . . . . 6 |- ( ( a e. NN \/ a = 0 ) -> ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
71	10, 70	sylbi 207	. . . . 5 |- ( a e. NN0 -> ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
72	71	com12 32	. . . 4 |- ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> ( a e. NN0 -> ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
73	72	ralrimiv 2965	. . 3 |- ( ( y e. NN /\ A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) ) -> A. a e. NN0 ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) )
74	73	ex 450	. 2 |- ( y e. NN -> ( A. x e. NN0 ( (#b ` x ) = y -> x = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) x ) x. ( 2 ^ k ) ) ) -> A. a e. NN0 ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )
75	9, 74	syl5bi 232	1 |- ( y e. NN -> ( A. a e. NN0 ( (#b ` a ) = y -> a = sum_ k e. ( 0..^ y ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) -> A. a e. NN0 ( (#b ` a ) = ( y + 1 ) -> a = sum_ k e. ( 0..^ ( y + 1 ) ) ( ( k (digit ` 2 ) a ) x. ( 2 ^ k ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   ^cexp 12860   sum_csu 14416  #bcblen 42363  digitcdig 42389

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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-logb 24503  df-blen 42364  df-dig 42390

This theorem is referenced by:  nn0sumshdiglem2  42416