Theorem dvgt0lem1 23765

Description: Lemma for dvgt0 23767 and dvlt0 23768. (Contributed by Mario Carneiro, 19-Feb-2015.)

Hypotheses

Ref	Expression
dvgt0.a	|- ( ph -> A e. RR )
dvgt0.b	|- ( ph -> B e. RR )
dvgt0.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
dvgt0lem.d	|- ( ph -> ( RR _D F ) : ( A (,) B ) --> S )

Assertion

Ref	Expression
dvgt0lem1	|- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( ( F ` Y ) - ( F ` X ) ) / ( Y - X ) ) e. S )

Proof of Theorem dvgt0lem1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . . . . . 7 |- ( A [,] B ) C_ RR*
2		simplrl 800	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> X e. ( A [,] B ) )
3	1, 2	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> X e. RR* )
4		simplrr 801	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> Y e. ( A [,] B ) )
5	1, 4	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> Y e. RR* )
6		dvgt0.a	. . . . . . . . . 10 |- ( ph -> A e. RR )
7		dvgt0.b	. . . . . . . . . 10 |- ( ph -> B e. RR )
8		iccssre 12255	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
9	6, 7, 8	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( A [,] B ) C_ RR )
10	9	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( A [,] B ) C_ RR )
11	10, 2	sseldd 3604	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> X e. RR )
12	10, 4	sseldd 3604	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> Y e. RR )
13		simpr 477	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> X < Y )
14	11, 12, 13	ltled 10185	. . . . . 6 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> X <_ Y )
15		ubicc2 12289	. . . . . 6 |- ( ( X e. RR* /\ Y e. RR* /\ X <_ Y ) -> Y e. ( X [,] Y ) )
16	3, 5, 14, 15	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> Y e. ( X [,] Y ) )
17		fvres 6207	. . . . 5 |- ( Y e. ( X [,] Y ) -> ( ( F |` ( X [,] Y ) ) ` Y ) = ( F ` Y ) )
18	16, 17	syl 17	. . . 4 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( F |` ( X [,] Y ) ) ` Y ) = ( F ` Y ) )
19		lbicc2 12288	. . . . . 6 |- ( ( X e. RR* /\ Y e. RR* /\ X <_ Y ) -> X e. ( X [,] Y ) )
20	3, 5, 14, 19	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> X e. ( X [,] Y ) )
21		fvres 6207	. . . . 5 |- ( X e. ( X [,] Y ) -> ( ( F |` ( X [,] Y ) ) ` X ) = ( F ` X ) )
22	20, 21	syl 17	. . . 4 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( F |` ( X [,] Y ) ) ` X ) = ( F ` X ) )
23	18, 22	oveq12d 6668	. . 3 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) = ( ( F ` Y ) - ( F ` X ) ) )
24	23	oveq1d 6665	. 2 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) = ( ( ( F ` Y ) - ( F ` X ) ) / ( Y - X ) ) )
25		iccss2 12244	. . . . . 6 |- ( ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) -> ( X [,] Y ) C_ ( A [,] B ) )
26	25	ad2antlr 763	. . . . 5 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( X [,] Y ) C_ ( A [,] B ) )
27		dvgt0.f	. . . . . 6 |- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
28	27	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> F e. ( ( A [,] B ) -cn-> RR ) )
29		rescncf 22700	. . . . 5 |- ( ( X [,] Y ) C_ ( A [,] B ) -> ( F e. ( ( A [,] B ) -cn-> RR ) -> ( F |` ( X [,] Y ) ) e. ( ( X [,] Y ) -cn-> RR ) ) )
30	26, 28, 29	sylc 65	. . . 4 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( F |` ( X [,] Y ) ) e. ( ( X [,] Y ) -cn-> RR ) )
31		dvgt0lem.d	. . . . . . . 8 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> S )
32	31	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( RR _D F ) : ( A (,) B ) --> S )
33	6	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> A e. RR )
34	33	rexrd 10089	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> A e. RR* )
35	7	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> B e. RR )
36		elicc2 12238	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
37	33, 35, 36	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
38	2, 37	mpbid 222	. . . . . . . . . 10 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( X e. RR /\ A <_ X /\ X <_ B ) )
39	38	simp2d 1074	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> A <_ X )
40		iooss1 12210	. . . . . . . . 9 |- ( ( A e. RR* /\ A <_ X ) -> ( X (,) Y ) C_ ( A (,) Y ) )
41	34, 39, 40	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( X (,) Y ) C_ ( A (,) Y ) )
42	35	rexrd 10089	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> B e. RR* )
43		elicc2 12238	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( Y e. ( A [,] B ) <-> ( Y e. RR /\ A <_ Y /\ Y <_ B ) ) )
44	33, 35, 43	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( Y e. ( A [,] B ) <-> ( Y e. RR /\ A <_ Y /\ Y <_ B ) ) )
45	4, 44	mpbid 222	. . . . . . . . . 10 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( Y e. RR /\ A <_ Y /\ Y <_ B ) )
46	45	simp3d 1075	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> Y <_ B )
47		iooss2 12211	. . . . . . . . 9 |- ( ( B e. RR* /\ Y <_ B ) -> ( A (,) Y ) C_ ( A (,) B ) )
48	42, 46, 47	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( A (,) Y ) C_ ( A (,) B ) )
49	41, 48	sstrd 3613	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( X (,) Y ) C_ ( A (,) B ) )
50	32, 49	fssresd 6071	. . . . . 6 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( RR _D F ) |` ( X (,) Y ) ) : ( X (,) Y ) --> S )
51		ax-resscn 9993	. . . . . . . . . 10 |- RR C_ CC
52	51	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> RR C_ CC )
53		cncff 22696	. . . . . . . . . . . 12 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
54	27, 53	syl 17	. . . . . . . . . . 11 |- ( ph -> F : ( A [,] B ) --> RR )
55	54	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> F : ( A [,] B ) --> RR )
56		fss 6056	. . . . . . . . . 10 |- ( ( F : ( A [,] B ) --> RR /\ RR C_ CC ) -> F : ( A [,] B ) --> CC )
57	55, 51, 56	sylancl 694	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> F : ( A [,] B ) --> CC )
58		iccssre 12255	. . . . . . . . . 10 |- ( ( X e. RR /\ Y e. RR ) -> ( X [,] Y ) C_ RR )
59	11, 12, 58	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( X [,] Y ) C_ RR )
60		eqid 2622	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
61	60	tgioo2 22606	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
62	60, 61	dvres 23675	. . . . . . . . 9 |- ( ( ( RR C_ CC /\ F : ( A [,] B ) --> CC ) /\ ( ( A [,] B ) C_ RR /\ ( X [,] Y ) C_ RR ) ) -> ( RR _D ( F |` ( X [,] Y ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( X [,] Y ) ) ) )
63	52, 57, 10, 59, 62	syl22anc 1327	. . . . . . . 8 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( RR _D ( F |` ( X [,] Y ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( X [,] Y ) ) ) )
64		iccntr 22624	. . . . . . . . . 10 |- ( ( X e. RR /\ Y e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( X [,] Y ) ) = ( X (,) Y ) )
65	11, 12, 64	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( X [,] Y ) ) = ( X (,) Y ) )
66	65	reseq2d 5396	. . . . . . . 8 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( X [,] Y ) ) ) = ( ( RR _D F ) |` ( X (,) Y ) ) )
67	63, 66	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( RR _D ( F |` ( X [,] Y ) ) ) = ( ( RR _D F ) |` ( X (,) Y ) ) )
68	67	feq1d 6030	. . . . . 6 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( RR _D ( F |` ( X [,] Y ) ) ) : ( X (,) Y ) --> S <-> ( ( RR _D F ) |` ( X (,) Y ) ) : ( X (,) Y ) --> S ) )
69	50, 68	mpbird 247	. . . . 5 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( RR _D ( F |` ( X [,] Y ) ) ) : ( X (,) Y ) --> S )
70		fdm 6051	. . . . 5 |- ( ( RR _D ( F |` ( X [,] Y ) ) ) : ( X (,) Y ) --> S -> dom ( RR _D ( F |` ( X [,] Y ) ) ) = ( X (,) Y ) )
71	69, 70	syl 17	. . . 4 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> dom ( RR _D ( F |` ( X [,] Y ) ) ) = ( X (,) Y ) )
72	11, 12, 13, 30, 71	mvth 23755	. . 3 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> E. z e. ( X (,) Y ) ( ( RR _D ( F |` ( X [,] Y ) ) ) ` z ) = ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) )
73	69	ffvelrnda 6359	. . . . 5 |- ( ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) /\ z e. ( X (,) Y ) ) -> ( ( RR _D ( F |` ( X [,] Y ) ) ) ` z ) e. S )
74		eleq1 2689	. . . . 5 |- ( ( ( RR _D ( F |` ( X [,] Y ) ) ) ` z ) = ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) -> ( ( ( RR _D ( F |` ( X [,] Y ) ) ) ` z ) e. S <-> ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) e. S ) )
75	73, 74	syl5ibcom 235	. . . 4 |- ( ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) /\ z e. ( X (,) Y ) ) -> ( ( ( RR _D ( F |` ( X [,] Y ) ) ) ` z ) = ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) -> ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) e. S ) )
76	75	rexlimdva 3031	. . 3 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( E. z e. ( X (,) Y ) ( ( RR _D ( F |` ( X [,] Y ) ) ) ` z ) = ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) -> ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) e. S ) )
77	72, 76	mpd 15	. 2 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( ( ( F |` ( X [,] Y ) ) ` Y ) - ( ( F |` ( X [,] Y ) ) ` X ) ) / ( Y - X ) ) e. S )
78	24, 77	eqeltrrd 2702	1 |- ( ( ( ph /\ ( X e. ( A [,] B ) /\ Y e. ( A [,] B ) ) ) /\ X < Y ) -> ( ( ( F ` Y ) - ( F ` X ) ) / ( Y - X ) ) e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

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-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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  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-cmp 21190  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

This theorem is referenced by:  dvgt0  23767  dvlt0  23768  dvge0  23769