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Theorem prdstmdd 21136
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.
StepHypRef 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 21088 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  Mnd )
65ssriv 3468 . . . 4  |- TopMnd  C_  Mnd
7 fss 5754 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  Mnd )  ->  R : I --> Mnd )
84, 6, 7sylancl 666 . . 3  |-  ( ph  ->  R : I --> Mnd )
91, 2, 3, 8prdsmndd 16568 . 2  |-  ( ph  ->  Y  e.  Mnd )
10 tmdtps 21089 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  TopSp )
1110ssriv 3468 . . . 4  |- TopMnd  C_  TopSp
12 fss 5754 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  TopSp )  ->  R : I --> TopSp )
134, 11, 12sylancl 666 . . 3  |-  ( ph  ->  R : I --> TopSp )
141, 3, 2, 13prdstps 20642 . 2  |-  ( ph  ->  Y  e.  TopSp )
15 eqid 2422 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
1633ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  S  e.  V
)
1723ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  I  e.  W
)
18 ffn 5746 . . . . . . . . 9  |-  ( R : I -->TopMnd  ->  R  Fn  I )
194, 18syl 17 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
20193ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  R  Fn  I
)
21 simp2 1006 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  f  e.  (
Base `  Y )
)
22 simp3 1007 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  g  e.  (
Base `  Y )
)
23 eqid 2422 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
241, 15, 16, 17, 20, 21, 22, 23prdsplusgval 15370 . . . . . 6  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  ( f ( +g  `  Y ) g )  =  ( k  e.  I  |->  ( ( f `  k
) ( +g  `  ( R `  k )
) ( g `  k ) ) ) )
2524mpt2eq3dva 6369 . . . . 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 2422 . . . . . 6  |-  ( +f `  Y )  =  ( +f `  Y )
2715, 23, 26plusffval 16492 . . . . 5  |-  ( +f `  Y )  =  ( f  e.  ( Base `  Y
) ,  g  e.  ( Base `  Y
)  |->  ( f ( +g  `  Y ) g ) )
28 vex 3083 . . . . . . . . . 10  |-  f  e. 
_V
29 vex 3083 . . . . . . . . . 10  |-  g  e. 
_V
3028, 29op1std 6817 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 1st `  z
)  =  f )
3130fveq1d 5883 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 1st `  z ) `  k
)  =  ( f `
 k ) )
3228, 29op2ndd 6818 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 2nd `  z
)  =  g )
3332fveq1d 5883 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 2nd `  z ) `  k
)  =  ( g `
 k ) )
3431, 33oveq12d 6323 . . . . . . 7  |-  ( z  =  <. f ,  g
>.  ->  ( ( ( 1st `  z ) `
 k ) ( +g  `  ( R `
 k ) ) ( ( 2nd `  z
) `  k )
)  =  ( ( f `  k ) ( +g  `  ( R `  k )
) ( g `  k ) ) )
3534mpteq2dv 4511 . . . . . 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 ) ) ) )
3635mpt2mpt 6402 . . . . 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
) ) ) )
3725, 27, 363eqtr4g 2488 . . . 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 2422 . . . . 5  |-  ( Xt_ `  ( TopOpen  o.  R )
)  =  ( Xt_ `  ( TopOpen  o.  R )
)
39 eqid 2422 . . . . . . . 8  |-  ( TopOpen `  Y )  =  (
TopOpen `  Y )
4015, 39istps 19949 . . . . . . 7  |-  ( Y  e.  TopSp 
<->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
4114, 40sylib 199 . . . . . 6  |-  ( ph  ->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
42 txtopon 20604 . . . . . 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
) ) ) )
4341, 41, 42syl2anc 665 . . . . 5  |-  ( ph  ->  ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  e.  (TopOn `  ( ( Base `  Y )  X.  ( Base `  Y
) ) ) )
44 topnfn 15323 . . . . . . . 8  |-  TopOpen  Fn  _V
45 ssv 3484 . . . . . . . 8  |-  TopSp  C_  _V
46 fnssres 5707 . . . . . . . 8  |-  ( (
TopOpen  Fn  _V  /\  TopSp  C_ 
_V )  ->  ( TopOpen  |`  TopSp )  Fn  TopSp )
4744, 45, 46mp2an 676 . . . . . . 7  |-  ( TopOpen  |`  TopSp
)  Fn  TopSp
48 fvres 5895 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  =  ( TopOpen `  x ) )
49 eqid 2422 . . . . . . . . . 10  |-  ( TopOpen `  x )  =  (
TopOpen `  x )
5049tpstop 19952 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( TopOpen `  x )  e.  Top )
5148, 50eqeltrd 2507 . . . . . . . 8  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
5251rgen 2781 . . . . . . 7  |-  A. x  e.  TopSp  ( ( TopOpen  |`  TopSp
) `  x )  e.  Top
53 ffnfv 6064 . . . . . . 7  |-  ( (
TopOpen 
|`  TopSp ) : TopSp --> Top  <->  ( ( TopOpen  |`  TopSp )  Fn  TopSp  /\ 
A. x  e.  TopSp  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
)
5447, 52, 53mpbir2an 928 . . . . . 6  |-  ( TopOpen  |`  TopSp
) : TopSp --> Top
55 fco2 5757 . . . . . 6  |-  ( ( ( TopOpen  |`  TopSp ) : TopSp --> Top 
/\  R : I -->
TopSp )  ->  ( TopOpen  o.  R ) : I --> Top )
5654, 13, 55sylancr 667 . . . . 5  |-  ( ph  ->  ( TopOpen  o.  R ) : I --> Top )
5734mpt2mpt 6402 . . . . . 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 2422 . . . . . . . 8  |-  ( TopOpen `  ( R `  k ) )  =  ( TopOpen `  ( R `  k ) )
59 eqid 2422 . . . . . . . 8  |-  ( +g  `  ( R `  k
) )  =  ( +g  `  ( R `
 k ) )
604ffvelrnda 6037 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( R `  k )  e. TopMnd )
6141adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  e.  (TopOn `  ( Base `  Y ) ) )
6261, 61cnmpt1st 20681 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  I )  ->  (
f  e.  ( Base `  Y ) ,  g  e.  ( Base `  Y
)  |->  f )  e.  ( ( ( TopOpen `  Y )  tX  ( TopOpen
`  Y ) )  Cn  ( TopOpen `  Y
) ) )
631, 3, 2, 19, 39prdstopn 20641 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( TopOpen `  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6463adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6564, 61eqeltrrd 2508 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  e.  (TopOn `  ( Base `  Y
) ) )
66 toponuni 19940 . . . . . . . . . . . . 13  |-  ( (
Xt_ `  ( TopOpen  o.  R
) )  e.  (TopOn `  ( Base `  Y
) )  ->  ( Base `  Y )  = 
U. ( Xt_ `  ( TopOpen  o.  R ) ) )
6765, 66syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  ( Base `  Y )  = 
U. ( Xt_ `  ( TopOpen  o.  R ) ) )
6867mpteq1d 4505 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  =  ( x  e. 
U. ( Xt_ `  ( TopOpen  o.  R ) ) 
|->  ( x `  k
) ) )
692adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  I  e.  W )
7056adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen  o.  R ) : I --> Top )
71 simpr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  k  e.  I )
72 eqid 2422 . . . . . . . . . . . . 13  |-  U. ( Xt_ `  ( TopOpen  o.  R
) )  =  U. ( Xt_ `  ( TopOpen  o.  R ) )
7372, 38ptpjcn 20624 . . . . . . . . . . . 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 ) ) )
7469, 70, 71, 73syl3anc 1264 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  U. ( Xt_ `  ( TopOpen  o.  R
) )  |->  ( x `
 k ) )  e.  ( ( Xt_ `  ( TopOpen  o.  R )
)  Cn  ( (
TopOpen  o.  R ) `  k ) ) )
7568, 74eqeltrd 2507 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( Xt_ `  ( TopOpen  o.  R )
)  Cn  ( (
TopOpen  o.  R ) `  k ) ) )
7664eqcomd 2430 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  =  (
TopOpen `  Y ) )
77 fvco3 5958 . . . . . . . . . . . 12  |-  ( ( R : I -->TopMnd  /\  k  e.  I )  ->  (
( TopOpen  o.  R ) `  k )  =  (
TopOpen `  ( R `  k ) ) )
784, 77sylan 473 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
( TopOpen  o.  R ) `  k )  =  (
TopOpen `  ( R `  k ) ) )
7976, 78oveq12d 6323 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
( Xt_ `  ( TopOpen  o.  R ) )  Cn  ( ( TopOpen  o.  R
) `  k )
)  =  ( (
TopOpen `  Y )  Cn  ( TopOpen `  ( R `  k ) ) ) )
8075, 79eleqtrd 2509 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( TopOpen `  Y )  Cn  ( TopOpen
`  ( R `  k ) ) ) )
81 fveq1 5880 . . . . . . . . 9  |-  ( x  =  f  ->  (
x `  k )  =  ( f `  k ) )
8261, 61, 62, 61, 80, 81cnmpt21 20684 . . . . . . . 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 )
) ) )
8361, 61cnmpt2nd 20682 . . . . . . . . 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 5880 . . . . . . . . 9  |-  ( x  =  g  ->  (
x `  k )  =  ( g `  k ) )
8561, 61, 83, 61, 80, 84cnmpt21 20684 . . . . . . . 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 )
) ) )
8658, 59, 60, 61, 61, 82, 85cnmpt2plusg 21101 . . . . . . 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 )
) ) )
8778oveq2d 6321 . . . . . . 7  |-  ( (
ph  /\  k  e.  I )  ->  (
( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( ( TopOpen  o.  R
) `  k )
)  =  ( ( ( TopOpen `  Y )  tX  ( TopOpen `  Y )
)  Cn  ( TopOpen `  ( R `  k ) ) ) )
8886, 87eleqtrrd 2510 . . . . . 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
) ) )
8957, 88syl5eqel 2511 . . . . 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 )
) )
9038, 43, 2, 56, 89ptcn 20640 . . . 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 ) ) ) )
9137, 90eqeltrd 2507 . . 3  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9263oveq2d 6321 . . 3  |-  ( ph  ->  ( ( ( TopOpen `  Y )  tX  ( TopOpen
`  Y ) )  Cn  ( TopOpen `  Y
) )  =  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9391, 92eleqtrrd 2510 . 2  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( TopOpen `  Y )
) )
9426, 39istmd 21087 . 2  |-  ( Y  e. TopMnd 
<->  ( Y  e.  Mnd  /\  Y  e.  TopSp  /\  ( +f `  Y
)  e.  ( ( ( TopOpen `  Y )  tX  ( TopOpen `  Y )
)  Cn  ( TopOpen `  Y ) ) ) )
959, 14, 93, 94syl3anbrc 1189 1  |-  ( ph  ->  Y  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   _Vcvv 3080    C_ wss 3436   <.cop 4004   U.cuni 4219    |-> cmpt 4482    X. cxp 4851    |` cres 4855    o. ccom 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   Basecbs 15120   +g cplusg 15189   TopOpenctopn 15319   Xt_cpt 15336   X_scprds 15343   +fcplusf 16484   Mndcmnd 16534   Topctop 19915  TopOnctopon 19916   TopSpctps 19917    Cn ccn 20238    tX ctx 20573  TopMndctmd 21083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fi 7934  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-hom 15213  df-cco 15214  df-rest 15320  df-topn 15321  df-0g 15339  df-topgen 15341  df-pt 15342  df-prds 15345  df-plusf 16486  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-top 19919  df-bases 19920  df-topon 19921  df-topsp 19922  df-cn 20241  df-cnp 20242  df-tx 20575  df-tmd 21085
This theorem is referenced by:  prdstgpd  21137
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