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Theorem prdstmdd 20352
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 20304 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  Mnd )
65ssriv 3503 . . . 4  |- TopMnd  C_  Mnd
7 fss 5732 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  Mnd )  ->  R : I --> Mnd )
84, 6, 7sylancl 662 . . 3  |-  ( ph  ->  R : I --> Mnd )
91, 2, 3, 8prdsmndd 15762 . 2  |-  ( ph  ->  Y  e.  Mnd )
10 tmdtps 20305 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  TopSp )
1110ssriv 3503 . . . 4  |- TopMnd  C_  TopSp
12 fss 5732 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  TopSp )  ->  R : I --> TopSp )
134, 11, 12sylancl 662 . . 3  |-  ( ph  ->  R : I --> TopSp )
141, 3, 2, 13prdstps 19860 . 2  |-  ( ph  ->  Y  e.  TopSp )
15 eqid 2462 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
1633ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  S  e.  V
)
1723ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  I  e.  W
)
18 ffn 5724 . . . . . . . . 9  |-  ( R : I -->TopMnd  ->  R  Fn  I )
194, 18syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
20193ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  R  Fn  I
)
21 simp2 992 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  f  e.  (
Base `  Y )
)
22 simp3 993 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  g  e.  (
Base `  Y )
)
23 eqid 2462 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
241, 15, 16, 17, 20, 21, 22, 23prdsplusgval 14719 . . . . . 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 6338 . . . . 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 2462 . . . . . 6  |-  ( +f `  Y )  =  ( +f `  Y )
2715, 23, 26plusffval 15735 . . . . 5  |-  ( +f `  Y )  =  ( f  e.  ( Base `  Y
) ,  g  e.  ( Base `  Y
)  |->  ( f ( +g  `  Y ) g ) )
28 vex 3111 . . . . . . . . . 10  |-  f  e. 
_V
29 vex 3111 . . . . . . . . . 10  |-  g  e. 
_V
3028, 29op1std 6786 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 1st `  z
)  =  f )
3130fveq1d 5861 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 1st `  z ) `  k
)  =  ( f `
 k ) )
3228, 29op2ndd 6787 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 2nd `  z
)  =  g )
3332fveq1d 5861 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 2nd `  z ) `  k
)  =  ( g `
 k ) )
3431, 33oveq12d 6295 . . . . . . 7  |-  ( z  =  <. f ,  g
>.  ->  ( ( ( 1st `  z ) `
 k ) ( +g  `  ( R `
 k ) ) ( ( 2nd `  z
) `  k )
)  =  ( ( f `  k ) ( +g  `  ( R `  k )
) ( g `  k ) ) )
3534mpteq2dv 4529 . . . . . 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 6371 . . . . 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 2528 . . . 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 2462 . . . . 5  |-  ( Xt_ `  ( TopOpen  o.  R )
)  =  ( Xt_ `  ( TopOpen  o.  R )
)
39 eqid 2462 . . . . . . . 8  |-  ( TopOpen `  Y )  =  (
TopOpen `  Y )
4015, 39istps 19199 . . . . . . 7  |-  ( Y  e.  TopSp 
<->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
4114, 40sylib 196 . . . . . 6  |-  ( ph  ->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
42 txtopon 19822 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  e.  (TopOn `  ( ( Base `  Y )  X.  ( Base `  Y
) ) ) )
44 topnfn 14672 . . . . . . . 8  |-  TopOpen  Fn  _V
45 ssv 3519 . . . . . . . 8  |-  TopSp  C_  _V
46 fnssres 5687 . . . . . . . 8  |-  ( (
TopOpen  Fn  _V  /\  TopSp  C_ 
_V )  ->  ( TopOpen  |`  TopSp )  Fn  TopSp )
4744, 45, 46mp2an 672 . . . . . . 7  |-  ( TopOpen  |`  TopSp
)  Fn  TopSp
48 fvres 5873 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  =  ( TopOpen `  x ) )
49 eqid 2462 . . . . . . . . . 10  |-  ( TopOpen `  x )  =  (
TopOpen `  x )
5049tpstop 19202 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( TopOpen `  x )  e.  Top )
5148, 50eqeltrd 2550 . . . . . . . 8  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
5251rgen 2819 . . . . . . 7  |-  A. x  e.  TopSp  ( ( TopOpen  |`  TopSp
) `  x )  e.  Top
53 ffnfv 6040 . . . . . . 7  |-  ( (
TopOpen 
|`  TopSp ) : TopSp --> Top  <->  ( ( TopOpen  |`  TopSp )  Fn  TopSp  /\ 
A. x  e.  TopSp  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
)
5447, 52, 53mpbir2an 913 . . . . . 6  |-  ( TopOpen  |`  TopSp
) : TopSp --> Top
55 fco2 5735 . . . . . 6  |-  ( ( ( TopOpen  |`  TopSp ) : TopSp --> Top 
/\  R : I -->
TopSp )  ->  ( TopOpen  o.  R ) : I --> Top )
5654, 13, 55sylancr 663 . . . . 5  |-  ( ph  ->  ( TopOpen  o.  R ) : I --> Top )
5734mpt2mpt 6371 . . . . . 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 2462 . . . . . . . 8  |-  ( TopOpen `  ( R `  k ) )  =  ( TopOpen `  ( R `  k ) )
59 eqid 2462 . . . . . . . 8  |-  ( +g  `  ( R `  k
) )  =  ( +g  `  ( R `
 k ) )
604ffvelrnda 6014 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( R `  k )  e. TopMnd )
6141adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  e.  (TopOn `  ( Base `  Y ) ) )
6261, 61cnmpt1st 19899 . . . . . . . . 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 19859 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( TopOpen `  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6463adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6564, 61eqeltrrd 2551 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  e.  (TopOn `  ( Base `  Y
) ) )
66 toponuni 19190 . . . . . . . . . . . . 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 4523 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  =  ( x  e. 
U. ( Xt_ `  ( TopOpen  o.  R ) ) 
|->  ( x `  k
) ) )
692adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  I  e.  W )
7056adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen  o.  R ) : I --> Top )
71 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  I )  ->  k  e.  I )
72 eqid 2462 . . . . . . . . . . . . 13  |-  U. ( Xt_ `  ( TopOpen  o.  R
) )  =  U. ( Xt_ `  ( TopOpen  o.  R ) )
7372, 38ptpjcn 19842 . . . . . . . . . . . 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 1223 . . . . . . . . . . 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 2550 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( Xt_ `  ( TopOpen  o.  R )
)  Cn  ( (
TopOpen  o.  R ) `  k ) ) )
7664eqcomd 2470 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  =  (
TopOpen `  Y ) )
77 fvco3 5937 . . . . . . . . . . . 12  |-  ( ( R : I -->TopMnd  /\  k  e.  I )  ->  (
( TopOpen  o.  R ) `  k )  =  (
TopOpen `  ( R `  k ) ) )
784, 77sylan 471 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  (
( TopOpen  o.  R ) `  k )  =  (
TopOpen `  ( R `  k ) ) )
7976, 78oveq12d 6295 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
( Xt_ `  ( TopOpen  o.  R ) )  Cn  ( ( TopOpen  o.  R
) `  k )
)  =  ( (
TopOpen `  Y )  Cn  ( TopOpen `  ( R `  k ) ) ) )
8075, 79eleqtrd 2552 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( TopOpen `  Y )  Cn  ( TopOpen
`  ( R `  k ) ) ) )
81 fveq1 5858 . . . . . . . . 9  |-  ( x  =  f  ->  (
x `  k )  =  ( f `  k ) )
8261, 61, 62, 61, 80, 81cnmpt21 19902 . . . . . . . 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 19900 . . . . . . . . 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 5858 . . . . . . . . 9  |-  ( x  =  g  ->  (
x `  k )  =  ( g `  k ) )
8561, 61, 83, 61, 80, 84cnmpt21 19902 . . . . . . . 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 20317 . . . . . . 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 6293 . . . . . . 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 2553 . . . . . 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 2554 . . . . 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 19858 . . . 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 2550 . . 3  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9263oveq2d 6293 . . 3  |-  ( ph  ->  ( ( ( TopOpen `  Y )  tX  ( TopOpen
`  Y ) )  Cn  ( TopOpen `  Y
) )  =  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9391, 92eleqtrrd 2553 . 2  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( TopOpen `  Y )
) )
9426, 39istmd 20303 . 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 1175 1  |-  ( ph  ->  Y  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    C_ wss 3471   <.cop 4028   U.cuni 4240    |-> cmpt 4500    X. cxp 4992    |` cres 4996    o. ccom 4998    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   Basecbs 14481   +g cplusg 14546   TopOpenctopn 14668   Xt_cpt 14685   X_scprds 14692   Mndcmnd 15717   +fcplusf 15720   Topctop 19156  TopOnctopon 19157   TopSpctps 19159    Cn ccn 19486    tX ctx 19791  TopMndctmd 20299
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fi 7862  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-topgen 14690  df-pt 14691  df-prds 14694  df-mnd 15723  df-plusf 15724  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cn 19489  df-cnp 19490  df-tx 19793  df-tmd 20301
This theorem is referenced by:  prdstgpd  20353
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