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Theorem prdstmdd 19653
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 19605 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  Mnd )
65ssriv 3357 . . . 4  |- TopMnd  C_  Mnd
7 fss 5564 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  Mnd )  ->  R : I --> Mnd )
84, 6, 7sylancl 657 . . 3  |-  ( ph  ->  R : I --> Mnd )
91, 2, 3, 8prdsmndd 15450 . 2  |-  ( ph  ->  Y  e.  Mnd )
10 tmdtps 19606 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  TopSp )
1110ssriv 3357 . . . 4  |- TopMnd  C_  TopSp
12 fss 5564 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  TopSp )  ->  R : I --> TopSp )
134, 11, 12sylancl 657 . . 3  |-  ( ph  ->  R : I --> TopSp )
141, 3, 2, 13prdstps 19161 . 2  |-  ( ph  ->  Y  e.  TopSp )
15 eqid 2441 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
1633ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  S  e.  V
)
1723ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  I  e.  W
)
18 ffn 5556 . . . . . . . . 9  |-  ( R : I -->TopMnd  ->  R  Fn  I )
194, 18syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
20193ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  R  Fn  I
)
21 simp2 984 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  f  e.  (
Base `  Y )
)
22 simp3 985 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  g  e.  (
Base `  Y )
)
23 eqid 2441 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
241, 15, 16, 17, 20, 21, 22, 23prdsplusgval 14407 . . . . . 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 6149 . . . . 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 2441 . . . . . 6  |-  ( +f `  Y )  =  ( +f `  Y )
2715, 23, 26plusffval 15423 . . . . 5  |-  ( +f `  Y )  =  ( f  e.  ( Base `  Y
) ,  g  e.  ( Base `  Y
)  |->  ( f ( +g  `  Y ) g ) )
28 vex 2973 . . . . . . . . . 10  |-  f  e. 
_V
29 vex 2973 . . . . . . . . . 10  |-  g  e. 
_V
3028, 29op1std 6586 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 1st `  z
)  =  f )
3130fveq1d 5690 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 1st `  z ) `  k
)  =  ( f `
 k ) )
3228, 29op2ndd 6587 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 2nd `  z
)  =  g )
3332fveq1d 5690 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 2nd `  z ) `  k
)  =  ( g `
 k ) )
3431, 33oveq12d 6108 . . . . . . 7  |-  ( z  =  <. f ,  g
>.  ->  ( ( ( 1st `  z ) `
 k ) ( +g  `  ( R `
 k ) ) ( ( 2nd `  z
) `  k )
)  =  ( ( f `  k ) ( +g  `  ( R `  k )
) ( g `  k ) ) )
3534mpteq2dv 4376 . . . . . 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 6181 . . . . 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 2498 . . . 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 2441 . . . . 5  |-  ( Xt_ `  ( TopOpen  o.  R )
)  =  ( Xt_ `  ( TopOpen  o.  R )
)
39 eqid 2441 . . . . . . . 8  |-  ( TopOpen `  Y )  =  (
TopOpen `  Y )
4015, 39istps 18500 . . . . . . 7  |-  ( Y  e.  TopSp 
<->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
4114, 40sylib 196 . . . . . 6  |-  ( ph  ->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
42 txtopon 19123 . . . . . 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 656 . . . . 5  |-  ( ph  ->  ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  e.  (TopOn `  ( ( Base `  Y )  X.  ( Base `  Y
) ) ) )
44 topnfn 14360 . . . . . . . 8  |-  TopOpen  Fn  _V
45 ssv 3373 . . . . . . . 8  |-  TopSp  C_  _V
46 fnssres 5521 . . . . . . . 8  |-  ( (
TopOpen  Fn  _V  /\  TopSp  C_ 
_V )  ->  ( TopOpen  |`  TopSp )  Fn  TopSp )
4744, 45, 46mp2an 667 . . . . . . 7  |-  ( TopOpen  |`  TopSp
)  Fn  TopSp
48 fvres 5701 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  =  ( TopOpen `  x ) )
49 eqid 2441 . . . . . . . . . 10  |-  ( TopOpen `  x )  =  (
TopOpen `  x )
5049tpstop 18503 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( TopOpen `  x )  e.  Top )
5148, 50eqeltrd 2515 . . . . . . . 8  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
5251rgen 2779 . . . . . . 7  |-  A. x  e.  TopSp  ( ( TopOpen  |`  TopSp
) `  x )  e.  Top
53 ffnfv 5866 . . . . . . 7  |-  ( (
TopOpen 
|`  TopSp ) : TopSp --> Top  <->  ( ( TopOpen  |`  TopSp )  Fn  TopSp  /\ 
A. x  e.  TopSp  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
)
5447, 52, 53mpbir2an 906 . . . . . 6  |-  ( TopOpen  |`  TopSp
) : TopSp --> Top
55 fco2 5566 . . . . . 6  |-  ( ( ( TopOpen  |`  TopSp ) : TopSp --> Top 
/\  R : I -->
TopSp )  ->  ( TopOpen  o.  R ) : I --> Top )
5654, 13, 55sylancr 658 . . . . 5  |-  ( ph  ->  ( TopOpen  o.  R ) : I --> Top )
5734mpt2mpt 6181 . . . . . 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 2441 . . . . . . . 8  |-  ( TopOpen `  ( R `  k ) )  =  ( TopOpen `  ( R `  k ) )
59 eqid 2441 . . . . . . . 8  |-  ( +g  `  ( R `  k
) )  =  ( +g  `  ( R `
 k ) )
604ffvelrnda 5840 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( R `  k )  e. TopMnd )
6141adantr 462 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  e.  (TopOn `  ( Base `  Y ) ) )
6261, 61cnmpt1st 19200 . . . . . . . . 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 19160 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( TopOpen `  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6463adantr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6564, 61eqeltrrd 2516 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  e.  (TopOn `  ( Base `  Y
) ) )
66 toponuni 18491 . . . . . . . . . . . . 13  |-  ( (
Xt_ `  ( TopOpen  o.  R
) )  e.  (TopOn `  ( Base `  Y
) )  ->  ( Base `  Y )  = 
U. ( Xt_ `  ( TopOpen  o.  R ) ) )
6765, 66syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  ( Base `  Y )  = 
U. ( Xt_ `  ( TopOpen  o.  R ) ) )
6867mpteq1d 4370 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  =  ( x  e. 
U. ( Xt_ `  ( TopOpen  o.  R ) ) 
|->  ( x `  k
) ) )
692adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  I  e.  W )
7056adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen  o.  R ) : I --> Top )
71 simpr 458 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  k  e.  I )
72 eqid 2441 . . . . . . . . . . . . 13  |-  U. ( Xt_ `  ( TopOpen  o.  R
) )  =  U. ( Xt_ `  ( TopOpen  o.  R ) )
7372, 38ptpjcn 19143 . . . . . . . . . . . 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 1213 . . . . . . . . . . 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 2515 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( Xt_ `  ( TopOpen  o.  R )
)  Cn  ( (
TopOpen  o.  R ) `  k ) ) )
7664eqcomd 2446 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  =  (
TopOpen `  Y ) )
77 fvco3 5765 . . . . . . . . . . . 12  |-  ( ( R : I -->TopMnd  /\  k  e.  I )  ->  (
( TopOpen  o.  R ) `  k )  =  (
TopOpen `  ( R `  k ) ) )
784, 77sylan 468 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
( TopOpen  o.  R ) `  k )  =  (
TopOpen `  ( R `  k ) ) )
7976, 78oveq12d 6108 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
( Xt_ `  ( TopOpen  o.  R ) )  Cn  ( ( TopOpen  o.  R
) `  k )
)  =  ( (
TopOpen `  Y )  Cn  ( TopOpen `  ( R `  k ) ) ) )
8075, 79eleqtrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( TopOpen `  Y )  Cn  ( TopOpen
`  ( R `  k ) ) ) )
81 fveq1 5687 . . . . . . . . 9  |-  ( x  =  f  ->  (
x `  k )  =  ( f `  k ) )
8261, 61, 62, 61, 80, 81cnmpt21 19203 . . . . . . . 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 19201 . . . . . . . . 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 5687 . . . . . . . . 9  |-  ( x  =  g  ->  (
x `  k )  =  ( g `  k ) )
8561, 61, 83, 61, 80, 84cnmpt21 19203 . . . . . . . 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 19618 . . . . . . 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 6106 . . . . . . 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 2518 . . . . . 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 2525 . . . . 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 19159 . . . 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 2515 . . 3  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9263oveq2d 6106 . . 3  |-  ( ph  ->  ( ( ( TopOpen `  Y )  tX  ( TopOpen
`  Y ) )  Cn  ( TopOpen `  Y
) )  =  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9391, 92eleqtrrd 2518 . 2  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( TopOpen `  Y )
) )
9426, 39istmd 19604 . 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 1167 1  |-  ( ph  ->  Y  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    C_ wss 3325   <.cop 3880   U.cuni 4088    e. cmpt 4347    X. cxp 4834    |` cres 4838    o. ccom 4840    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   +g cplusg 14234   TopOpenctopn 14356   Xt_cpt 14373   X_scprds 14380   Mndcmnd 15405   +fcplusf 15408   Topctop 18457  TopOnctopon 18458   TopSpctps 18460    Cn ccn 18787    tX ctx 19092  TopMndctmd 19600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-topgen 14378  df-pt 14379  df-prds 14382  df-mnd 15411  df-plusf 15412  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cn 18790  df-cnp 18791  df-tx 19094  df-tmd 19602
This theorem is referenced by:  prdstgpd  19654
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