Theorem dvcmulf 23708

Description: The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Hypotheses

Ref	Expression
dvcmul.s	|- ( ph -> S e. { RR , CC } )
dvcmul.f	|- ( ph -> F : X --> CC )
dvcmul.a	|- ( ph -> A e. CC )
dvcmulf.df	|- ( ph -> dom ( S _D F ) = X )

Assertion

Ref	Expression
dvcmulf	|- ( ph -> ( S _D ( ( S X. { A } ) oF x. F ) ) = ( ( S X. { A } ) oF x. ( S _D F ) ) )

Proof of Theorem dvcmulf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvcmul.s	. . 3 |- ( ph -> S e. { RR , CC } )
2		dvcmul.a	. . . . 5 |- ( ph -> A e. CC )
3		fconstg 6092	. . . . 5 |- ( A e. CC -> ( X X. { A } ) : X --> { A } )
4	2, 3	syl 17	. . . 4 |- ( ph -> ( X X. { A } ) : X --> { A } )
5	2	snssd 4340	. . . 4 |- ( ph -> { A } C_ CC )
6	4, 5	fssd 6057	. . 3 |- ( ph -> ( X X. { A } ) : X --> CC )
7		dvcmul.f	. . 3 |- ( ph -> F : X --> CC )
8		c0ex 10034	. . . . . 6 |- 0 e. _V
9	8	fconst 6091	. . . . 5 |- ( X X. { 0 } ) : X --> { 0 }
10		recnprss 23668	. . . . . . . . 9 |- ( S e. { RR , CC } -> S C_ CC )
11	1, 10	syl 17	. . . . . . . 8 |- ( ph -> S C_ CC )
12		fconstg 6092	. . . . . . . . . 10 |- ( A e. CC -> ( S X. { A } ) : S --> { A } )
13	2, 12	syl 17	. . . . . . . . 9 |- ( ph -> ( S X. { A } ) : S --> { A } )
14	13, 5	fssd 6057	. . . . . . . 8 |- ( ph -> ( S X. { A } ) : S --> CC )
15		ssid 3624	. . . . . . . . 9 |- S C_ S
16	15	a1i 11	. . . . . . . 8 |- ( ph -> S C_ S )
17		dvcmulf.df	. . . . . . . . 9 |- ( ph -> dom ( S _D F ) = X )
18		dvbsss 23666	. . . . . . . . . 10 |- dom ( S _D F ) C_ S
19	18	a1i 11	. . . . . . . . 9 |- ( ph -> dom ( S _D F ) C_ S )
20	17, 19	eqsstr3d 3640	. . . . . . . 8 |- ( ph -> X C_ S )
21		eqid 2622	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
22		eqid 2622	. . . . . . . . 9 |- ( ( TopOpen ` ℂfld ) ↾t S ) = ( ( TopOpen ` ℂfld ) ↾t S )
23	21, 22	dvres 23675	. . . . . . . 8 |- ( ( ( S C_ CC /\ ( S X. { A } ) : S --> CC ) /\ ( S C_ S /\ X C_ S ) ) -> ( S _D ( ( S X. { A } ) |` X ) ) = ( ( S _D ( S X. { A } ) ) |` ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) ) )
24	11, 14, 16, 20, 23	syl22anc 1327	. . . . . . 7 |- ( ph -> ( S _D ( ( S X. { A } ) |` X ) ) = ( ( S _D ( S X. { A } ) ) |` ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) ) )
25	20	resmptd 5452	. . . . . . . . 9 |- ( ph -> ( ( x e. S |-> A ) |` X ) = ( x e. X |-> A ) )
26		fconstmpt 5163	. . . . . . . . . 10 |- ( S X. { A } ) = ( x e. S |-> A )
27	26	reseq1i 5392	. . . . . . . . 9 |- ( ( S X. { A } ) |` X ) = ( ( x e. S |-> A ) |` X )
28		fconstmpt 5163	. . . . . . . . 9 |- ( X X. { A } ) = ( x e. X |-> A )
29	25, 27, 28	3eqtr4g 2681	. . . . . . . 8 |- ( ph -> ( ( S X. { A } ) |` X ) = ( X X. { A } ) )
30	29	oveq2d 6666	. . . . . . 7 |- ( ph -> ( S _D ( ( S X. { A } ) |` X ) ) = ( S _D ( X X. { A } ) ) )
31	20	resmptd 5452	. . . . . . . 8 |- ( ph -> ( ( x e. S |-> 0 ) |` X ) = ( x e. X |-> 0 ) )
32		fconstg 6092	. . . . . . . . . . . . . 14 |- ( A e. CC -> ( CC X. { A } ) : CC --> { A } )
33	2, 32	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( CC X. { A } ) : CC --> { A } )
34	33, 5	fssd 6057	. . . . . . . . . . . 12 |- ( ph -> ( CC X. { A } ) : CC --> CC )
35		ssid 3624	. . . . . . . . . . . . 13 |- CC C_ CC
36	35	a1i 11	. . . . . . . . . . . 12 |- ( ph -> CC C_ CC )
37		dvconst 23680	. . . . . . . . . . . . . . . 16 |- ( A e. CC -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
38	2, 37	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
39	38	dmeqd 5326	. . . . . . . . . . . . . 14 |- ( ph -> dom ( CC _D ( CC X. { A } ) ) = dom ( CC X. { 0 } ) )
40	8	fconst 6091	. . . . . . . . . . . . . . 15 |- ( CC X. { 0 } ) : CC --> { 0 }
41	40	fdmi 6052	. . . . . . . . . . . . . 14 |- dom ( CC X. { 0 } ) = CC
42	39, 41	syl6eq 2672	. . . . . . . . . . . . 13 |- ( ph -> dom ( CC _D ( CC X. { A } ) ) = CC )
43	11, 42	sseqtr4d 3642	. . . . . . . . . . . 12 |- ( ph -> S C_ dom ( CC _D ( CC X. { A } ) ) )
44		dvres3 23677	. . . . . . . . . . . 12 |- ( ( ( S e. { RR , CC } /\ ( CC X. { A } ) : CC --> CC ) /\ ( CC C_ CC /\ S C_ dom ( CC _D ( CC X. { A } ) ) ) ) -> ( S _D ( ( CC X. { A } ) |` S ) ) = ( ( CC _D ( CC X. { A } ) ) |` S ) )
45	1, 34, 36, 43, 44	syl22anc 1327	. . . . . . . . . . 11 |- ( ph -> ( S _D ( ( CC X. { A } ) |` S ) ) = ( ( CC _D ( CC X. { A } ) ) |` S ) )
46		xpssres 5434	. . . . . . . . . . . . 13 |- ( S C_ CC -> ( ( CC X. { A } ) |` S ) = ( S X. { A } ) )
47	11, 46	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( CC X. { A } ) |` S ) = ( S X. { A } ) )
48	47	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( S _D ( ( CC X. { A } ) |` S ) ) = ( S _D ( S X. { A } ) ) )
49	38	reseq1d 5395	. . . . . . . . . . . 12 |- ( ph -> ( ( CC _D ( CC X. { A } ) ) |` S ) = ( ( CC X. { 0 } ) |` S ) )
50		xpssres 5434	. . . . . . . . . . . . 13 |- ( S C_ CC -> ( ( CC X. { 0 } ) |` S ) = ( S X. { 0 } ) )
51	11, 50	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( CC X. { 0 } ) |` S ) = ( S X. { 0 } ) )
52	49, 51	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( ( CC _D ( CC X. { A } ) ) |` S ) = ( S X. { 0 } ) )
53	45, 48, 52	3eqtr3d 2664	. . . . . . . . . 10 |- ( ph -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
54		fconstmpt 5163	. . . . . . . . . 10 |- ( S X. { 0 } ) = ( x e. S |-> 0 )
55	53, 54	syl6eq 2672	. . . . . . . . 9 |- ( ph -> ( S _D ( S X. { A } ) ) = ( x e. S |-> 0 ) )
56	21	cnfldtopon 22586	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
57		resttopon 20965	. . . . . . . . . . . . 13 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ S C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t S ) e. (TopOn ` S ) )
58	56, 11, 57	sylancr 695	. . . . . . . . . . . 12 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t S ) e. (TopOn ` S ) )
59		topontop 20718	. . . . . . . . . . . 12 |- ( ( ( TopOpen ` ℂfld ) ↾t S ) e. (TopOn ` S ) -> ( ( TopOpen ` ℂfld ) ↾t S ) e. Top )
60	58, 59	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t S ) e. Top )
61		toponuni 20719	. . . . . . . . . . . . 13 |- ( ( ( TopOpen ` ℂfld ) ↾t S ) e. (TopOn ` S ) -> S = U. ( ( TopOpen ` ℂfld ) ↾t S ) )
62	58, 61	syl 17	. . . . . . . . . . . 12 |- ( ph -> S = U. ( ( TopOpen ` ℂfld ) ↾t S ) )
63	20, 62	sseqtrd 3641	. . . . . . . . . . 11 |- ( ph -> X C_ U. ( ( TopOpen ` ℂfld ) ↾t S ) )
64		eqid 2622	. . . . . . . . . . . 12 |- U. ( ( TopOpen ` ℂfld ) ↾t S ) = U. ( ( TopOpen ` ℂfld ) ↾t S )
65	64	ntrss2 20861	. . . . . . . . . . 11 |- ( ( ( ( TopOpen ` ℂfld ) ↾t S ) e. Top /\ X C_ U. ( ( TopOpen ` ℂfld ) ↾t S ) ) -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) C_ X )
66	60, 63, 65	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) C_ X )
67	11, 7, 20, 22, 21	dvbssntr 23664	. . . . . . . . . . 11 |- ( ph -> dom ( S _D F ) C_ ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) )
68	17, 67	eqsstr3d 3640	. . . . . . . . . 10 |- ( ph -> X C_ ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) )
69	66, 68	eqssd 3620	. . . . . . . . 9 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) = X )
70	55, 69	reseq12d 5397	. . . . . . . 8 |- ( ph -> ( ( S _D ( S X. { A } ) ) |` ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) ) = ( ( x e. S |-> 0 ) |` X ) )
71		fconstmpt 5163	. . . . . . . . 9 |- ( X X. { 0 } ) = ( x e. X |-> 0 )
72	71	a1i 11	. . . . . . . 8 |- ( ph -> ( X X. { 0 } ) = ( x e. X |-> 0 ) )
73	31, 70, 72	3eqtr4d 2666	. . . . . . 7 |- ( ph -> ( ( S _D ( S X. { A } ) ) |` ( ( int ` ( ( TopOpen ` ℂfld ) ↾t S ) ) ` X ) ) = ( X X. { 0 } ) )
74	24, 30, 73	3eqtr3d 2664	. . . . . 6 |- ( ph -> ( S _D ( X X. { A } ) ) = ( X X. { 0 } ) )
75	74	feq1d 6030	. . . . 5 |- ( ph -> ( ( S _D ( X X. { A } ) ) : X --> { 0 } <-> ( X X. { 0 } ) : X --> { 0 } ) )
76	9, 75	mpbiri 248	. . . 4 |- ( ph -> ( S _D ( X X. { A } ) ) : X --> { 0 } )
77		fdm 6051	. . . 4 |- ( ( S _D ( X X. { A } ) ) : X --> { 0 } -> dom ( S _D ( X X. { A } ) ) = X )
78	76, 77	syl 17	. . 3 |- ( ph -> dom ( S _D ( X X. { A } ) ) = X )
79	1, 6, 7, 78, 17	dvmulf 23706	. 2 |- ( ph -> ( S _D ( ( X X. { A } ) oF x. F ) ) = ( ( ( S _D ( X X. { A } ) ) oF x. F ) oF + ( ( S _D F ) oF x. ( X X. { A } ) ) ) )
80		sseqin2 3817	. . . . . 6 |- ( X C_ S <-> ( S i^i X ) = X )
81	20, 80	sylib 208	. . . . 5 |- ( ph -> ( S i^i X ) = X )
82	81	mpteq1d 4738	. . . 4 |- ( ph -> ( x e. ( S i^i X ) |-> ( A x. ( F ` x ) ) ) = ( x e. X |-> ( A x. ( F ` x ) ) ) )
83		ffn 6045	. . . . . 6 |- ( ( S X. { A } ) : S --> { A } -> ( S X. { A } ) Fn S )
84	13, 83	syl 17	. . . . 5 |- ( ph -> ( S X. { A } ) Fn S )
85		ffn 6045	. . . . . 6 |- ( F : X --> CC -> F Fn X )
86	7, 85	syl 17	. . . . 5 |- ( ph -> F Fn X )
87	1, 20	ssexd 4805	. . . . 5 |- ( ph -> X e. _V )
88		eqid 2622	. . . . 5 |- ( S i^i X ) = ( S i^i X )
89		fvconst2g 6467	. . . . . 6 |- ( ( A e. CC /\ x e. S ) -> ( ( S X. { A } ) ` x ) = A )
90	2, 89	sylan 488	. . . . 5 |- ( ( ph /\ x e. S ) -> ( ( S X. { A } ) ` x ) = A )
91		eqidd 2623	. . . . 5 |- ( ( ph /\ x e. X ) -> ( F ` x ) = ( F ` x ) )
92	84, 86, 1, 87, 88, 90, 91	offval 6904	. . . 4 |- ( ph -> ( ( S X. { A } ) oF x. F ) = ( x e. ( S i^i X ) |-> ( A x. ( F ` x ) ) ) )
93		ffn 6045	. . . . . 6 |- ( ( X X. { A } ) : X --> { A } -> ( X X. { A } ) Fn X )
94	4, 93	syl 17	. . . . 5 |- ( ph -> ( X X. { A } ) Fn X )
95		inidm 3822	. . . . 5 |- ( X i^i X ) = X
96		fvconst2g 6467	. . . . . 6 |- ( ( A e. CC /\ x e. X ) -> ( ( X X. { A } ) ` x ) = A )
97	2, 96	sylan 488	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( X X. { A } ) ` x ) = A )
98	94, 86, 87, 87, 95, 97, 91	offval 6904	. . . 4 |- ( ph -> ( ( X X. { A } ) oF x. F ) = ( x e. X |-> ( A x. ( F ` x ) ) ) )
99	82, 92, 98	3eqtr4d 2666	. . 3 |- ( ph -> ( ( S X. { A } ) oF x. F ) = ( ( X X. { A } ) oF x. F ) )
100	99	oveq2d 6666	. 2 |- ( ph -> ( S _D ( ( S X. { A } ) oF x. F ) ) = ( S _D ( ( X X. { A } ) oF x. F ) ) )
101	81	mpteq1d 4738	. . 3 |- ( ph -> ( x e. ( S i^i X ) |-> ( A x. ( ( S _D F ) ` x ) ) ) = ( x e. X |-> ( A x. ( ( S _D F ) ` x ) ) ) )
102		dvfg 23670	. . . . . . 7 |- ( S e. { RR , CC } -> ( S _D F ) : dom ( S _D F ) --> CC )
103	1, 102	syl 17	. . . . . 6 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
104	17	feq2d 6031	. . . . . 6 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
105	103, 104	mpbid 222	. . . . 5 |- ( ph -> ( S _D F ) : X --> CC )
106		ffn 6045	. . . . 5 |- ( ( S _D F ) : X --> CC -> ( S _D F ) Fn X )
107	105, 106	syl 17	. . . 4 |- ( ph -> ( S _D F ) Fn X )
108		eqidd 2623	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( S _D F ) ` x ) )
109	84, 107, 1, 87, 88, 90, 108	offval 6904	. . 3 |- ( ph -> ( ( S X. { A } ) oF x. ( S _D F ) ) = ( x e. ( S i^i X ) |-> ( A x. ( ( S _D F ) ` x ) ) ) )
110		0cnd 10033	. . . . 5 |- ( ( ph /\ x e. X ) -> 0 e. CC )
111		ovexd 6680	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. A ) e. _V )
112	74	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( S _D ( X X. { A } ) ) oF x. F ) = ( ( X X. { 0 } ) oF x. F ) )
113		0cnd 10033	. . . . . . . 8 |- ( ph -> 0 e. CC )
114		mul02 10214	. . . . . . . . 9 |- ( x e. CC -> ( 0 x. x ) = 0 )
115	114	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( 0 x. x ) = 0 )
116	87, 7, 113, 113, 115	caofid2 6928	. . . . . . 7 |- ( ph -> ( ( X X. { 0 } ) oF x. F ) = ( X X. { 0 } ) )
117	112, 116	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( S _D ( X X. { A } ) ) oF x. F ) = ( X X. { 0 } ) )
118	117, 71	syl6eq 2672	. . . . 5 |- ( ph -> ( ( S _D ( X X. { A } ) ) oF x. F ) = ( x e. X |-> 0 ) )
119		fvexd 6203	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. _V )
120	2	adantr 481	. . . . . 6 |- ( ( ph /\ x e. X ) -> A e. CC )
121	105	feqmptd 6249	. . . . . 6 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
122	28	a1i 11	. . . . . 6 |- ( ph -> ( X X. { A } ) = ( x e. X |-> A ) )
123	87, 119, 120, 121, 122	offval2 6914	. . . . 5 |- ( ph -> ( ( S _D F ) oF x. ( X X. { A } ) ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. A ) ) )
124	87, 110, 111, 118, 123	offval2 6914	. . . 4 |- ( ph -> ( ( ( S _D ( X X. { A } ) ) oF x. F ) oF + ( ( S _D F ) oF x. ( X X. { A } ) ) ) = ( x e. X |-> ( 0 + ( ( ( S _D F ) ` x ) x. A ) ) ) )
125	105	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. CC )
126	125, 120	mulcld 10060	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. A ) e. CC )
127	126	addid2d 10237	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( 0 + ( ( ( S _D F ) ` x ) x. A ) ) = ( ( ( S _D F ) ` x ) x. A ) )
128	125, 120	mulcomd 10061	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. A ) = ( A x. ( ( S _D F ) ` x ) ) )
129	127, 128	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. X ) -> ( 0 + ( ( ( S _D F ) ` x ) x. A ) ) = ( A x. ( ( S _D F ) ` x ) ) )
130	129	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. X |-> ( 0 + ( ( ( S _D F ) ` x ) x. A ) ) ) = ( x e. X |-> ( A x. ( ( S _D F ) ` x ) ) ) )
131	124, 130	eqtrd 2656	. . 3 |- ( ph -> ( ( ( S _D ( X X. { A } ) ) oF x. F ) oF + ( ( S _D F ) oF x. ( X X. { A } ) ) ) = ( x e. X |-> ( A x. ( ( S _D F ) ` x ) ) ) )
132	101, 109, 131	3eqtr4d 2666	. 2 |- ( ph -> ( ( S X. { A } ) oF x. ( S _D F ) ) = ( ( ( S _D ( X X. { A } ) ) oF x. F ) oF + ( ( S _D F ) oF x. ( X X. { A } ) ) ) )
133	79, 100, 132	3eqtr4d 2666	1 |- ( ph -> ( S _D ( ( S X. { A } ) oF x. F ) ) = ( ( S X. { A } ) oF x. ( S _D F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   {cpr 4179   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   intcnt 20821   _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-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-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:  dvsinax  40127