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Theorem prdstmdd 19810
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 19762 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  Mnd )
65ssriv 3458 . . . 4  |- TopMnd  C_  Mnd
7 fss 5665 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  Mnd )  ->  R : I --> Mnd )
84, 6, 7sylancl 662 . . 3  |-  ( ph  ->  R : I --> Mnd )
91, 2, 3, 8prdsmndd 15556 . 2  |-  ( ph  ->  Y  e.  Mnd )
10 tmdtps 19763 . . . . 5  |-  ( x  e. TopMnd  ->  x  e.  TopSp )
1110ssriv 3458 . . . 4  |- TopMnd  C_  TopSp
12 fss 5665 . . . 4  |-  ( ( R : I -->TopMnd  /\ TopMnd  C_  TopSp )  ->  R : I --> TopSp )
134, 11, 12sylancl 662 . . 3  |-  ( ph  ->  R : I --> TopSp )
141, 3, 2, 13prdstps 19318 . 2  |-  ( ph  ->  Y  e.  TopSp )
15 eqid 2451 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
1633ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  S  e.  V
)
1723ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  I  e.  W
)
18 ffn 5657 . . . . . . . . 9  |-  ( R : I -->TopMnd  ->  R  Fn  I )
194, 18syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
20193ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  R  Fn  I
)
21 simp2 989 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  f  e.  (
Base `  Y )
)
22 simp3 990 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( Base `  Y )  /\  g  e.  ( Base `  Y ) )  ->  g  e.  (
Base `  Y )
)
23 eqid 2451 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
241, 15, 16, 17, 20, 21, 22, 23prdsplusgval 14513 . . . . . 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 6249 . . . . 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 2451 . . . . . 6  |-  ( +f `  Y )  =  ( +f `  Y )
2715, 23, 26plusffval 15529 . . . . 5  |-  ( +f `  Y )  =  ( f  e.  ( Base `  Y
) ,  g  e.  ( Base `  Y
)  |->  ( f ( +g  `  Y ) g ) )
28 vex 3071 . . . . . . . . . 10  |-  f  e. 
_V
29 vex 3071 . . . . . . . . . 10  |-  g  e. 
_V
3028, 29op1std 6687 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 1st `  z
)  =  f )
3130fveq1d 5791 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 1st `  z ) `  k
)  =  ( f `
 k ) )
3228, 29op2ndd 6688 . . . . . . . . 9  |-  ( z  =  <. f ,  g
>.  ->  ( 2nd `  z
)  =  g )
3332fveq1d 5791 . . . . . . . 8  |-  ( z  =  <. f ,  g
>.  ->  ( ( 2nd `  z ) `  k
)  =  ( g `
 k ) )
3431, 33oveq12d 6208 . . . . . . 7  |-  ( z  =  <. f ,  g
>.  ->  ( ( ( 1st `  z ) `
 k ) ( +g  `  ( R `
 k ) ) ( ( 2nd `  z
) `  k )
)  =  ( ( f `  k ) ( +g  `  ( R `  k )
) ( g `  k ) ) )
3534mpteq2dv 4477 . . . . . 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 6282 . . . . 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 2517 . . . 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 2451 . . . . 5  |-  ( Xt_ `  ( TopOpen  o.  R )
)  =  ( Xt_ `  ( TopOpen  o.  R )
)
39 eqid 2451 . . . . . . . 8  |-  ( TopOpen `  Y )  =  (
TopOpen `  Y )
4015, 39istps 18657 . . . . . . 7  |-  ( Y  e.  TopSp 
<->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
4114, 40sylib 196 . . . . . 6  |-  ( ph  ->  ( TopOpen `  Y )  e.  (TopOn `  ( Base `  Y ) ) )
42 txtopon 19280 . . . . . 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 14466 . . . . . . . 8  |-  TopOpen  Fn  _V
45 ssv 3474 . . . . . . . 8  |-  TopSp  C_  _V
46 fnssres 5622 . . . . . . . 8  |-  ( (
TopOpen  Fn  _V  /\  TopSp  C_ 
_V )  ->  ( TopOpen  |`  TopSp )  Fn  TopSp )
4744, 45, 46mp2an 672 . . . . . . 7  |-  ( TopOpen  |`  TopSp
)  Fn  TopSp
48 fvres 5803 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  =  ( TopOpen `  x ) )
49 eqid 2451 . . . . . . . . . 10  |-  ( TopOpen `  x )  =  (
TopOpen `  x )
5049tpstop 18660 . . . . . . . . 9  |-  ( x  e.  TopSp  ->  ( TopOpen `  x )  e.  Top )
5148, 50eqeltrd 2539 . . . . . . . 8  |-  ( x  e.  TopSp  ->  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
5251rgen 2889 . . . . . . 7  |-  A. x  e.  TopSp  ( ( TopOpen  |`  TopSp
) `  x )  e.  Top
53 ffnfv 5968 . . . . . . 7  |-  ( (
TopOpen 
|`  TopSp ) : TopSp --> Top  <->  ( ( TopOpen  |`  TopSp )  Fn  TopSp  /\ 
A. x  e.  TopSp  ( ( TopOpen  |`  TopSp ) `  x
)  e.  Top )
)
5447, 52, 53mpbir2an 911 . . . . . 6  |-  ( TopOpen  |`  TopSp
) : TopSp --> Top
55 fco2 5667 . . . . . 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 6282 . . . . . 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 2451 . . . . . . . 8  |-  ( TopOpen `  ( R `  k ) )  =  ( TopOpen `  ( R `  k ) )
59 eqid 2451 . . . . . . . 8  |-  ( +g  `  ( R `  k
) )  =  ( +g  `  ( R `
 k ) )
604ffvelrnda 5942 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( R `  k )  e. TopMnd )
6141adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  e.  (TopOn `  ( Base `  Y ) ) )
6261, 61cnmpt1st 19357 . . . . . . . . 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 19317 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( TopOpen `  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6463adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  I )  ->  ( TopOpen
`  Y )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
6564, 61eqeltrrd 2540 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  e.  (TopOn `  ( Base `  Y
) ) )
66 toponuni 18648 . . . . . . . . . . . . 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 4471 . . . . . . . . . . 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 2451 . . . . . . . . . . . . 13  |-  U. ( Xt_ `  ( TopOpen  o.  R
) )  =  U. ( Xt_ `  ( TopOpen  o.  R ) )
7372, 38ptpjcn 19300 . . . . . . . . . . . 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 1219 . . . . . . . . . . 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 2539 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( Xt_ `  ( TopOpen  o.  R )
)  Cn  ( (
TopOpen  o.  R ) `  k ) ) )
7664eqcomd 2459 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  I )  ->  ( Xt_ `  ( TopOpen  o.  R
) )  =  (
TopOpen `  Y ) )
77 fvco3 5867 . . . . . . . . . . . 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 6208 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  I )  ->  (
( Xt_ `  ( TopOpen  o.  R ) )  Cn  ( ( TopOpen  o.  R
) `  k )
)  =  ( (
TopOpen `  Y )  Cn  ( TopOpen `  ( R `  k ) ) ) )
8075, 79eleqtrd 2541 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  I )  ->  (
x  e.  ( Base `  Y )  |->  ( x `
 k ) )  e.  ( ( TopOpen `  Y )  Cn  ( TopOpen
`  ( R `  k ) ) ) )
81 fveq1 5788 . . . . . . . . 9  |-  ( x  =  f  ->  (
x `  k )  =  ( f `  k ) )
8261, 61, 62, 61, 80, 81cnmpt21 19360 . . . . . . . 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 19358 . . . . . . . . 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 5788 . . . . . . . . 9  |-  ( x  =  g  ->  (
x `  k )  =  ( g `  k ) )
8561, 61, 83, 61, 80, 84cnmpt21 19360 . . . . . . . 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 19775 . . . . . . 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 6206 . . . . . . 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 2542 . . . . . 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 2543 . . . . 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 19316 . . . 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 2539 . . 3  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9263oveq2d 6206 . . 3  |-  ( ph  ->  ( ( ( TopOpen `  Y )  tX  ( TopOpen
`  Y ) )  Cn  ( TopOpen `  Y
) )  =  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( Xt_ `  ( TopOpen  o.  R ) ) ) )
9391, 92eleqtrrd 2542 . 2  |-  ( ph  ->  ( +f `  Y )  e.  ( ( ( TopOpen `  Y
)  tX  ( TopOpen `  Y ) )  Cn  ( TopOpen `  Y )
) )
9426, 39istmd 19761 . 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 1172 1  |-  ( ph  ->  Y  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3068    C_ wss 3426   <.cop 3981   U.cuni 4189    |-> cmpt 4448    X. cxp 4936    |` cres 4940    o. ccom 4942    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   1stc1st 6675   2ndc2nd 6676   Basecbs 14276   +g cplusg 14340   TopOpenctopn 14462   Xt_cpt 14479   X_scprds 14486   Mndcmnd 15511   +fcplusf 15514   Topctop 18614  TopOnctopon 18615   TopSpctps 18617    Cn ccn 18944    tX ctx 19249  TopMndctmd 19757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fi 7762  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-topgen 14484  df-pt 14485  df-prds 14488  df-mnd 15517  df-plusf 15518  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cn 18947  df-cnp 18948  df-tx 19251  df-tmd 19759
This theorem is referenced by:  prdstgpd  19811
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