Theorem prdstmdd 21927

Description: The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
prdstmdd.y	|- Y = ( S X_s R )
prdstmdd.i	|- ( ph -> I e. W )
prdstmdd.s	|- ( ph -> S e. V )
prdstmdd.r	|- ( ph -> R : I -->TopMnd )

Assertion

Ref	Expression
prdstmdd	|- ( ph -> Y e. TopMnd )

Proof of Theorem prdstmdd

Dummy variables f g k x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdstmdd.y	. . 3 |- Y = ( S X_s R )
2		prdstmdd.i	. . 3 |- ( ph -> I e. W )
3		prdstmdd.s	. . 3 |- ( ph -> S e. V )
4		prdstmdd.r	. . . 4 |- ( ph -> R : I -->TopMnd )
5		tmdmnd 21879	. . . . 5 |- ( x e. TopMnd -> x e. Mnd )
6	5	ssriv 3607	. . . 4 |- TopMnd C_ Mnd
7		fss 6056	. . . 4 |- ( ( R : I -->TopMnd /\ TopMnd C_ Mnd ) -> R : I --> Mnd )
8	4, 6, 7	sylancl 694	. . 3 |- ( ph -> R : I --> Mnd )
9	1, 2, 3, 8	prdsmndd 17323	. 2 |- ( ph -> Y e. Mnd )
10		tmdtps 21880	. . . . 5 |- ( x e. TopMnd -> x e. TopSp )
11	10	ssriv 3607	. . . 4 |- TopMnd C_ TopSp
12		fss 6056	. . . 4 |- ( ( R : I -->TopMnd /\ TopMnd C_ TopSp ) -> R : I --> TopSp )
13	4, 11, 12	sylancl 694	. . 3 |- ( ph -> R : I --> TopSp )
14	1, 3, 2, 13	prdstps 21432	. 2 |- ( ph -> Y e. TopSp )
15		eqid 2622	. . . . . . 7 |- ( Base ` Y ) = ( Base ` Y )
16	3	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> S e. V )
17	2	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> I e. W )
18		ffn 6045	. . . . . . . . 9 |- ( R : I -->TopMnd -> R Fn I )
19	4, 18	syl 17	. . . . . . . 8 |- ( ph -> R Fn I )
20	19	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> R Fn I )
21		simp2 1062	. . . . . . 7 |- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> f e. ( Base ` Y ) )
22		simp3 1063	. . . . . . 7 |- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> g e. ( Base ` Y ) )
23		eqid 2622	. . . . . . 7 |- ( +g ` Y ) = ( +g ` Y )
24	1, 15, 16, 17, 20, 21, 22, 23	prdsplusgval 16133	. . . . . 6 |- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> ( f ( +g ` Y ) g ) = ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) )
25	24	mpt2eq3dva 6719	. . . . 5 |- ( ph -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( f ( +g ` Y ) g ) ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) ) )
26		eqid 2622	. . . . . 6 |- ( +f ` Y ) = ( +f ` Y )
27	15, 23, 26	plusffval 17247	. . . . 5 |- ( +f ` Y ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( f ( +g ` Y ) g ) )
28		vex 3203	. . . . . . . . . 10 |- f e. _V
29		vex 3203	. . . . . . . . . 10 |- g e. _V
30	28, 29	op1std 7178	. . . . . . . . 9 |- ( z = <. f , g >. -> ( 1st ` z ) = f )
31	30	fveq1d 6193	. . . . . . . 8 |- ( z = <. f , g >. -> ( ( 1st ` z ) ` k ) = ( f ` k ) )
32	28, 29	op2ndd 7179	. . . . . . . . 9 |- ( z = <. f , g >. -> ( 2nd ` z ) = g )
33	32	fveq1d 6193	. . . . . . . 8 |- ( z = <. f , g >. -> ( ( 2nd ` z ) ` k ) = ( g ` k ) )
34	31, 33	oveq12d 6668	. . . . . . 7 |- ( z = <. f , g >. -> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) = ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) )
35	34	mpteq2dv 4745	. . . . . 6 |- ( z = <. f , g >. -> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) = ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) )
36	35	mpt2mpt 6752	. . . . 5 |- ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) )
37	25, 27, 36	3eqtr4g 2681	. . . 4 |- ( ph -> ( +f ` Y ) = ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) ) )
38		eqid 2622	. . . . 5 |- ( Xt_ ` ( TopOpen o. R ) ) = ( Xt_ ` ( TopOpen o. R ) )
39		eqid 2622	. . . . . . . 8 |- ( TopOpen ` Y ) = ( TopOpen ` Y )
40	15, 39	istps 20738	. . . . . . 7 |- ( Y e. TopSp <-> ( TopOpen ` Y ) e. (TopOn ` ( Base ` Y ) ) )
41	14, 40	sylib 208	. . . . . 6 |- ( ph -> ( TopOpen ` Y ) e. (TopOn ` ( Base ` Y ) ) )
42		txtopon 21394	. . . . . 6 |- ( ( ( TopOpen ` Y ) e. (TopOn ` ( Base ` Y ) ) /\ ( TopOpen ` Y ) e. (TopOn ` ( Base ` Y ) ) ) -> ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) e. (TopOn ` ( ( Base ` Y ) X. ( Base ` Y ) ) ) )
43	41, 41, 42	syl2anc 693	. . . . 5 |- ( ph -> ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) e. (TopOn ` ( ( Base ` Y ) X. ( Base ` Y ) ) ) )
44		topnfn 16086	. . . . . . . 8 |- TopOpen Fn _V
45		ssv 3625	. . . . . . . 8 |- TopSp C_ _V
46		fnssres 6004	. . . . . . . 8 |- ( ( TopOpen Fn _V /\ TopSp C_ _V ) -> ( TopOpen |` TopSp ) Fn TopSp )
47	44, 45, 46	mp2an 708	. . . . . . 7 |- ( TopOpen |` TopSp ) Fn TopSp
48		fvres 6207	. . . . . . . . 9 |- ( x e. TopSp -> ( ( TopOpen |` TopSp ) ` x ) = ( TopOpen ` x ) )
49		eqid 2622	. . . . . . . . . 10 |- ( TopOpen ` x ) = ( TopOpen ` x )
50	49	tpstop 20741	. . . . . . . . 9 |- ( x e. TopSp -> ( TopOpen ` x ) e. Top )
51	48, 50	eqeltrd 2701	. . . . . . . 8 |- ( x e. TopSp -> ( ( TopOpen |` TopSp ) ` x ) e. Top )
52	51	rgen 2922	. . . . . . 7 |- A. x e. TopSp ( ( TopOpen |` TopSp ) ` x ) e. Top
53		ffnfv 6388	. . . . . . 7 |- ( ( TopOpen |` TopSp ) : TopSp --> Top <-> ( ( TopOpen |` TopSp ) Fn TopSp /\ A. x e. TopSp ( ( TopOpen |` TopSp ) ` x ) e. Top ) )
54	47, 52, 53	mpbir2an 955	. . . . . 6 |- ( TopOpen |` TopSp ) : TopSp --> Top
55		fco2 6059	. . . . . 6 |- ( ( ( TopOpen |` TopSp ) : TopSp --> Top /\ R : I --> TopSp ) -> ( TopOpen o. R ) : I --> Top )
56	54, 13, 55	sylancr 695	. . . . 5 |- ( ph -> ( TopOpen o. R ) : I --> Top )
57	34	mpt2mpt 6752	. . . . . 6 |- ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) )
58		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ( R ` k ) ) = ( TopOpen ` ( R ` k ) )
59		eqid 2622	. . . . . . . 8 |- ( +g ` ( R ` k ) ) = ( +g ` ( R ` k ) )
60	4	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ k e. I ) -> ( R ` k ) e. TopMnd )
61	41	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. I ) -> ( TopOpen ` Y ) e. (TopOn ` ( Base ` Y ) ) )
62	61, 61	cnmpt1st 21471	. . . . . . . . 9 |- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> f ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) )
63	1, 3, 2, 19, 39	prdstopn 21431	. . . . . . . . . . . . . . 15 |- ( ph -> ( TopOpen ` Y ) = ( Xt_ ` ( TopOpen o. R ) ) )
64	63	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. I ) -> ( TopOpen ` Y ) = ( Xt_ ` ( TopOpen o. R ) ) )
65	64, 61	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. I ) -> ( Xt_ ` ( TopOpen o. R ) ) e. (TopOn ` ( Base ` Y ) ) )
66		toponuni 20719	. . . . . . . . . . . . 13 |- ( ( Xt_ ` ( TopOpen o. R ) ) e. (TopOn ` ( Base ` Y ) ) -> ( Base ` Y ) = U. ( Xt_ ` ( TopOpen o. R ) ) )
67	65, 66	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ k e. I ) -> ( Base ` Y ) = U. ( Xt_ ` ( TopOpen o. R ) ) )
68	67	mpteq1d 4738	. . . . . . . . . . 11 |- ( ( ph /\ k e. I ) -> ( x e. ( Base ` Y ) |-> ( x ` k ) ) = ( x e. U. ( Xt_ ` ( TopOpen o. R ) ) |-> ( x ` k ) ) )
69	2	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. I ) -> I e. W )
70	56	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. I ) -> ( TopOpen o. R ) : I --> Top )
71		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ k e. I ) -> k e. I )
72		eqid 2622	. . . . . . . . . . . . 13 |- U. ( Xt_ ` ( TopOpen o. R ) ) = U. ( Xt_ ` ( TopOpen o. R ) )
73	72, 38	ptpjcn 21414	. . . . . . . . . . . 12 |- ( ( I e. W /\ ( TopOpen o. R ) : I --> Top /\ k e. I ) -> ( x e. U. ( Xt_ ` ( TopOpen o. R ) ) |-> ( x ` k ) ) e. ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) )
74	69, 70, 71, 73	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ k e. I ) -> ( x e. U. ( Xt_ ` ( TopOpen o. R ) ) |-> ( x ` k ) ) e. ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) )
75	68, 74	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ k e. I ) -> ( x e. ( Base ` Y ) |-> ( x ` k ) ) e. ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) )
76	64	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ph /\ k e. I ) -> ( Xt_ ` ( TopOpen o. R ) ) = ( TopOpen ` Y ) )
77		fvco3 6275	. . . . . . . . . . . 12 |- ( ( R : I -->TopMnd /\ k e. I ) -> ( ( TopOpen o. R ) ` k ) = ( TopOpen ` ( R ` k ) ) )
78	4, 77	sylan 488	. . . . . . . . . . 11 |- ( ( ph /\ k e. I ) -> ( ( TopOpen o. R ) ` k ) = ( TopOpen ` ( R ` k ) ) )
79	76, 78	oveq12d 6668	. . . . . . . . . 10 |- ( ( ph /\ k e. I ) -> ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) = ( ( TopOpen ` Y ) Cn ( TopOpen ` ( R ` k ) ) ) )
80	75, 79	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ k e. I ) -> ( x e. ( Base ` Y ) |-> ( x ` k ) ) e. ( ( TopOpen ` Y ) Cn ( TopOpen ` ( R ` k ) ) ) )
81		fveq1 6190	. . . . . . . . 9 |- ( x = f -> ( x ` k ) = ( f ` k ) )
82	61, 61, 62, 61, 80, 81	cnmpt21 21474	. . . . . . . 8 |- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( f ` k ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) )
83	61, 61	cnmpt2nd 21472	. . . . . . . . 9 |- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> g ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) )
84		fveq1 6190	. . . . . . . . 9 |- ( x = g -> ( x ` k ) = ( g ` k ) )
85	61, 61, 83, 61, 80, 84	cnmpt21 21474	. . . . . . . 8 |- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( g ` k ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) )
86	58, 59, 60, 61, 61, 82, 85	cnmpt2plusg 21892	. . . . . . 7 |- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) )
87	78	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ k e. I ) -> ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( ( TopOpen o. R ) ` k ) ) = ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) )
88	86, 87	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( ( TopOpen o. R ) ` k ) ) )
89	57, 88	syl5eqel 2705	. . . . 5 |- ( ( ph /\ k e. I ) -> ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( ( TopOpen o. R ) ` k ) ) )
90	38, 43, 2, 56, 89	ptcn 21430	. . . 4 |- ( ph -> ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( Xt_ ` ( TopOpen o. R ) ) ) )
91	37, 90	eqeltrd 2701	. . 3 |- ( ph -> ( +f ` Y ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( Xt_ ` ( TopOpen o. R ) ) ) )
92	63	oveq2d 6666	. . 3 |- ( ph -> ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) = ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( Xt_ ` ( TopOpen o. R ) ) ) )
93	91, 92	eleqtrrd 2704	. 2 |- ( ph -> ( +f ` Y ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) )
94	26, 39	istmd 21878	. 2 |- ( Y e. TopMnd <-> ( Y e. Mnd /\ Y e. TopSp /\ ( +f ` Y ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) ) )
95	9, 14, 93, 94	syl3anbrc 1246	1 |- ( ph -> Y e. TopMnd )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   Xt_cpt 16099   X__scprds 16106   +fcplusf 17239   Mndcmnd 17294   Topctop 20698  TopOnctopon 20715   TopSpctps 20736   Cn ccn 21028   tX ctx 21363  TopMndctmd 21874

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-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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  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-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-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-topgen 16104  df-pt 16105  df-prds 16108  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-tmd 21876

This theorem is referenced by:  prdstgpd  21928