ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xmettxlem Unicode version

Theorem xmettxlem 13676
Description: Lemma for xmettx 13677. (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 13674 . . . . . . . 8  |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
5 blrn 13579 . . . . . . . 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 13611 . . . . . . . . . . . . . 14  |-  ( M  e.  ( *Met `  X )  ->  J  e.  Top )
102, 9syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
11 xmettx.k . . . . . . . . . . . . . . 15  |-  K  =  ( MetOpen `  N )
1211mopntop 13611 . . . . . . . . . . . . . 14  |-  ( N  e.  ( *Met `  Y )  ->  K  e.  Top )
133, 12syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
14 mpoexga 6207 . . . . . . . . . . . . 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 4888 . . . . . . . . . . . 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 13228 . . . . . . . . . 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 6160 . . . . . . . . . . . . 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 13657 . . . . . . . . . . . 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 6161 . . . . . . . . . . . . 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 13657 . . . . . . . . . . . 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 13675 . . . . . . . . . . . 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 2210 . . . . . . . . . . 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 4637 . . . . . . . . . . . . 13  |-  ( r  =  ( ( 1st `  z ) ( ball `  M ) p )  ->  ( r  X.  s )  =  ( ( ( 1st `  z
) ( ball `  M
) p )  X.  s ) )
3736eqeq2d 2189 . . . . . . . . . . . 12  |-  ( r  =  ( ( 1st `  z ) ( ball `  M ) p )  ->  ( w  =  ( r  X.  s
)  <->  w  =  (
( ( 1st `  z
) ( ball `  M
) p )  X.  s ) ) )
38 xpeq2 4638 . . . . . . . . . . . . 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 2189 . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 2177 . . . . . . . . . . . 12  |-  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )  =  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) )
4342elrnmpog 5981 . . . . . . . . . . 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 2741 . . . . . . . . . 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 3156 . . . . . . . 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 2602 . . . . . 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 3161 . . 3  |-  ( ph  ->  ran  ( ball `  P
)  C_  ( topGen ` 
ran  ( r  e.  J ,  s  e.  K  |->  ( r  X.  s ) ) ) )
52 blex 13554 . . . . 5  |-  ( P  e.  ( *Met `  ( X  X.  Y
) )  ->  ( ball `  P )  e. 
_V )
53 rnexg 4888 . . . . 5  |-  ( (
ball `  P )  e.  _V  ->  ran  ( ball `  P )  e.  _V )
544, 52, 533syl 17 . . . 4  |-  ( ph  ->  ran  ( ball `  P
)  e.  _V )
55 tgss3 13245 . . . 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 13609 . . 3  |-  ( P  e.  ( *Met `  ( X  X.  Y
) )  ->  L  =  ( topGen `  ran  ( ball `  P )
) )
604, 59syl 14 . 2  |-  ( ph  ->  L  =  ( topGen ` 
ran  ( ball `  P
) ) )
61 eqid 2177 . . . 4  |-  ran  (
r  e.  J , 
s  e.  K  |->  ( r  X.  s ) )  =  ran  (
r  e.  J , 
s  e.  K  |->  ( r  X.  s ) )
6261txval 13422 . . 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 3200 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 2148   E.wrex 2456   _Vcvv 2737    C_ wss 3129   {cpr 3592    X. cxp 4621   ran crn 4624   ` cfv 5212  (class class class)co 5869    e. cmpo 5871   1stc1st 6133   2ndc2nd 6134   supcsup 6975   RR*cxr 7981    < clt 7982   topGenctg 12651   *Metcxmet 13147   ballcbl 13149   MetOpencmopn 13152   Topctop 13162    tX ctx 13419
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-map 6644  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-tx 13420
This theorem is referenced by:  xmettx  13677
  Copyright terms: Public domain W3C validator