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Theorem xmettxlem 13580
Description: Lemma for xmettx 13581. (Contributed by Jim Kingdon, 15-Oct-2023.)
Hypotheses
Ref Expression
xmetxp.p  |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )
)
xmetxp.1  |-  ( ph  ->  M  e.  ( *Met `  X ) )
xmetxp.2  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
xmettx.j  |-  J  =  ( MetOpen `  M )
xmettx.k  |-  K  =  ( MetOpen `  N )
xmettx.l  |-  L  =  ( MetOpen `  P )
Assertion
Ref Expression
xmettxlem  |-  ( ph  ->  L  C_  ( J  tX  K ) )
Distinct variable groups:    u, M, v   
u, N, v    u, X, v    u, Y, v
Allowed substitution hints:    ph( v, u)    P( v, u)    J( v, u)    K( v, u)    L( v, u)

Proof of Theorem xmettxlem
Dummy variables  p  r  s  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetxp.p . . . . . . . . 9  |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )
)
2 xmetxp.1 . . . . . . . . 9  |-  ( ph  ->  M  e.  ( *Met `  X ) )
3 xmetxp.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
41, 2, 3xmetxp 13578 . . . . . . . 8  |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
5 blrn 13483 . . . . . . . 8  |-  ( P  e.  ( *Met `  ( X  X.  Y
) )  ->  (
w  e.  ran  ( ball `  P )  <->  E. z  e.  ( X  X.  Y
) E. p  e. 
RR*  w  =  ( z ( ball `  P
) p ) ) )
64, 5syl 14 . . . . . . 7  |-  ( ph  ->  ( w  e.  ran  ( ball `  P )  <->  E. z  e.  ( X  X.  Y ) E. p  e.  RR*  w  =  ( z (
ball `  P )
p ) ) )
76biimpa 296 . . . . . 6  |-  ( (
ph  /\  w  e.  ran  ( ball `  P
) )  ->  E. z  e.  ( X  X.  Y
) E. p  e. 
RR*  w  =  ( z ( ball `  P
) p ) )
8 xmettx.j . . . . . . . . . . . . . . 15  |-  J  =  ( MetOpen `  M )
98mopntop 13515 . . . . . . . . . . . . . 14  |-  ( M  e.  ( *Met `  X )  ->  J  e.  Top )
102, 9syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
11 xmettx.k . . . . . . . . . . . . . . 15  |-  K  =  ( MetOpen `  N )
1211mopntop 13515 . . . . . . . . . . . . . 14  |-  ( N  e.  ( *Met `  Y )  ->  K  e.  Top )
133, 12syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
14 mpoexga 6203 . . . . . . . . . . . . 13  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  e.  _V )
1510, 13, 14syl2anc 411 . . . . . . . . . . . 12  |-  ( ph  ->  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  e.  _V )
16 rnexg 4885 . . . . . . . . . . . 12  |-  ( ( r  e.  J , 
s  e.  K  |->  ( r  X.  s ) )  e.  _V  ->  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  e.  _V )
1715, 16syl 14 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )  e. 
_V )
1817ad3antrrr 492 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  e.  _V )
19 bastg 13132 . . . . . . . . . 10  |-  ( ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  e.  _V  ->  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )  C_  ( topGen `  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
2018, 19syl 14 . . . . . . . . 9  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  C_  ( topGen `
 ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
212ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  M  e.  ( *Met `  X ) )
22 simplrl 535 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  -> 
z  e.  ( X  X.  Y ) )
23 xp1st 6156 . . . . . . . . . . . . 13  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
2422, 23syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  -> 
( 1st `  z
)  e.  X )
25 simplrr 536 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  p  e.  RR* )
268blopn 13561 . . . . . . . . . . . 12  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  z
)  e.  X  /\  p  e.  RR* )  -> 
( ( 1st `  z
) ( ball `  M
) p )  e.  J )
2721, 24, 25, 26syl3anc 1238 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  -> 
( ( 1st `  z
) ( ball `  M
) p )  e.  J )
283ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  N  e.  ( *Met `  Y ) )
29 xp2nd 6157 . . . . . . . . . . . . 13  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
3022, 29syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  -> 
( 2nd `  z
)  e.  Y )
3111blopn 13561 . . . . . . . . . . . 12  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  z
)  e.  Y  /\  p  e.  RR* )  -> 
( ( 2nd `  z
) ( ball `  N
) p )  e.  K )
3228, 30, 25, 31syl3anc 1238 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  -> 
( ( 2nd `  z
) ( ball `  N
) p )  e.  K )
33 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  w  =  ( z
( ball `  P )
p ) )
341, 21, 28, 25, 22xmetxpbl 13579 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  -> 
( z ( ball `  P ) p )  =  ( ( ( 1st `  z ) ( ball `  M
) p )  X.  ( ( 2nd `  z
) ( ball `  N
) p ) ) )
3533, 34eqtrd 2208 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  w  =  ( (
( 1st `  z
) ( ball `  M
) p )  X.  ( ( 2nd `  z
) ( ball `  N
) p ) ) )
36 xpeq1 4634 . . . . . . . . . . . . 13  |-  ( r  =  ( ( 1st `  z ) ( ball `  M ) p )  ->  ( r  X.  s )  =  ( ( ( 1st `  z
) ( ball `  M
) p )  X.  s ) )
3736eqeq2d 2187 . . . . . . . . . . . 12  |-  ( r  =  ( ( 1st `  z ) ( ball `  M ) p )  ->  ( w  =  ( r  X.  s
)  <->  w  =  (
( ( 1st `  z
) ( ball `  M
) p )  X.  s ) ) )
38 xpeq2 4635 . . . . . . . . . . . . 13  |-  ( s  =  ( ( 2nd `  z ) ( ball `  N ) p )  ->  ( ( ( 1st `  z ) ( ball `  M
) p )  X.  s )  =  ( ( ( 1st `  z
) ( ball `  M
) p )  X.  ( ( 2nd `  z
) ( ball `  N
) p ) ) )
3938eqeq2d 2187 . . . . . . . . . . . 12  |-  ( s  =  ( ( 2nd `  z ) ( ball `  N ) p )  ->  ( w  =  ( ( ( 1st `  z ) ( ball `  M ) p )  X.  s )  <->  w  =  ( ( ( 1st `  z ) ( ball `  M ) p )  X.  ( ( 2nd `  z ) ( ball `  N ) p ) ) ) )
4037, 39rspc2ev 2854 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  z
) ( ball `  M
) p )  e.  J  /\  ( ( 2nd `  z ) ( ball `  N
) p )  e.  K  /\  w  =  ( ( ( 1st `  z ) ( ball `  M ) p )  X.  ( ( 2nd `  z ) ( ball `  N ) p ) ) )  ->  E. r  e.  J  E. s  e.  K  w  =  ( r  X.  s
) )
4127, 32, 35, 40syl3anc 1238 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  E. r  e.  J  E. s  e.  K  w  =  ( r  X.  s ) )
42 eqid 2175 . . . . . . . . . . . 12  |-  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )  =  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )
4342elrnmpog 5977 . . . . . . . . . . 11  |-  ( w  e.  _V  ->  (
w  e.  ran  (
r  e.  J , 
s  e.  K  |->  ( r  X.  s ) )  <->  E. r  e.  J  E. s  e.  K  w  =  ( r  X.  s ) ) )
4443elv 2739 . . . . . . . . . 10  |-  ( w  e.  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )  <->  E. r  e.  J  E. s  e.  K  w  =  ( r  X.  s ) )
4541, 44sylibr 134 . . . . . . . . 9  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  w  e.  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) )
4620, 45sseldd 3154 . . . . . . . 8  |-  ( ( ( ( ph  /\  w  e.  ran  ( ball `  P ) )  /\  ( z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  /\  w  =  ( z (
ball `  P )
p ) )  ->  w  e.  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
4746ex 115 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  ran  ( ball `  P
) )  /\  (
z  e.  ( X  X.  Y )  /\  p  e.  RR* ) )  ->  ( w  =  ( z ( ball `  P ) p )  ->  w  e.  (
topGen `  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) ) )
4847rexlimdvva 2600 . . . . . 6  |-  ( (
ph  /\  w  e.  ran  ( ball `  P
) )  ->  ( E. z  e.  ( X  X.  Y ) E. p  e.  RR*  w  =  ( z (
ball `  P )
p )  ->  w  e.  ( topGen `  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) ) )
497, 48mpd 13 . . . . 5  |-  ( (
ph  /\  w  e.  ran  ( ball `  P
) )  ->  w  e.  ( topGen `  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
5049ex 115 . . . 4  |-  ( ph  ->  ( w  e.  ran  ( ball `  P )  ->  w  e.  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) ) )
5150ssrdv 3159 . . 3  |-  ( ph  ->  ran  ( ball `  P
)  C_  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
52 blex 13458 . . . . 5  |-  ( P  e.  ( *Met `  ( X  X.  Y
) )  ->  ( ball `  P )  e. 
_V )
53 rnexg 4885 . . . . 5  |-  ( (
ball `  P )  e.  _V  ->  ran  ( ball `  P )  e.  _V )
544, 52, 533syl 17 . . . 4  |-  ( ph  ->  ran  ( ball `  P
)  e.  _V )
55 tgss3 13149 . . . 4  |-  ( ( ran  ( ball `  P
)  e.  _V  /\  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s
) )  e.  _V )  ->  ( ( topGen ` 
ran  ( ball `  P
) )  C_  ( topGen `
 ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) )  <->  ran  ( ball `  P
)  C_  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) ) )
5654, 17, 55syl2anc 411 . . 3  |-  ( ph  ->  ( ( topGen `  ran  ( ball `  P )
)  C_  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) )  <->  ran  ( ball `  P
)  C_  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) ) )
5751, 56mpbird 167 . 2  |-  ( ph  ->  ( topGen `  ran  ( ball `  P ) )  C_  ( topGen `  ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
58 xmettx.l . . . 4  |-  L  =  ( MetOpen `  P )
5958mopnval 13513 . . 3  |-  ( P  e.  ( *Met `  ( X  X.  Y
) )  ->  L  =  ( topGen `  ran  ( ball `  P )
) )
604, 59syl 14 . 2  |-  ( ph  ->  L  =  ( topGen ` 
ran  ( ball `  P
) ) )
61 eqid 2175 . . . 4  |-  ran  (
r  e.  J , 
s  e.  K  |->  ( r  X.  s ) )  =  ran  (
r  e.  J , 
s  e.  K  |->  ( r  X.  s ) )
6261txval 13326 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  tX  K
)  =  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
6310, 13, 62syl2anc 411 . 2  |-  ( ph  ->  ( J  tX  K
)  =  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
6457, 60, 633sstr4d 3198 1  |-  ( ph  ->  L  C_  ( J  tX  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2146   E.wrex 2454   _Vcvv 2735    C_ wss 3127   {cpr 3590    X. cxp 4618   ran crn 4621   ` cfv 5208  (class class class)co 5865    e. cmpo 5867   1stc1st 6129   2ndc2nd 6130   supcsup 6971   RR*cxr 7965    < clt 7966   topGenctg 12625   *Metcxmet 13051   ballcbl 13053   MetOpencmopn 13056   Topctop 13066    tX ctx 13323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-xneg 9743  df-xadd 9744  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976  df-topgen 12631  df-psmet 13058  df-xmet 13059  df-bl 13061  df-mopn 13062  df-top 13067  df-topon 13080  df-bases 13112  df-tx 13324
This theorem is referenced by:  xmettx  13581
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