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Theorem xpsxmetlem 21472
Description: Lemma for xpsxmet 21473. (Contributed by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
xpsds.t  |-  T  =  ( R  X.s  S )
xpsds.x  |-  X  =  ( Base `  R
)
xpsds.y  |-  Y  =  ( Base `  S
)
xpsds.1  |-  ( ph  ->  R  e.  V )
xpsds.2  |-  ( ph  ->  S  e.  W )
xpsds.p  |-  P  =  ( dist `  T
)
xpsds.m  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
xpsds.n  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
xpsds.3  |-  ( ph  ->  M  e.  ( *Met `  X ) )
xpsds.4  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
Assertion
Ref Expression
xpsxmetlem  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
Distinct variable groups:    x, y    x, M    x, N    ph, x    x, R    x, S    x, X, y    x, Y, y   
x, W
Allowed substitution hints:    ph( y)    P( x, y)    R( y)    S( y)    T( x, y)    M( y)    N( y)    V( x, y)    W( y)

Proof of Theorem xpsxmetlem
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . 3  |-  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
2 eqid 2471 . . 3  |-  ( Base `  ( (Scalar `  R
) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
3 eqid 2471 . . 3  |-  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )
4 eqid 2471 . . 3  |-  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  |`  (
( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) ) )
5 eqid 2471 . . 3  |-  ( dist `  ( (Scalar `  R
) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )  =  ( dist `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
6 fvex 5889 . . . 4  |-  (Scalar `  R )  e.  _V
76a1i 11 . . 3  |-  ( ph  ->  (Scalar `  R )  e.  _V )
8 2on 7208 . . . 4  |-  2o  e.  On
98a1i 11 . . 3  |-  ( ph  ->  2o  e.  On )
10 fvex 5889 . . . 4  |-  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V
1110a1i 11 . . 3  |-  ( (
ph  /\  k  e.  2o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  e. 
_V )
12 elpri 3976 . . . . 5  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
13 df2o3 7213 . . . . 5  |-  2o  =  { (/) ,  1o }
1412, 13eleq2s 2567 . . . 4  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
15 xpsds.3 . . . . . . 7  |-  ( ph  ->  M  e.  ( *Met `  X ) )
1615adantr 472 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  M  e.  ( *Met `  X
) )
17 fveq2 5879 . . . . . . . . . 10  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
18 xpsds.1 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  V )
19 xpsc0 15544 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
2018, 19syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
2117, 20sylan9eqr 2527 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( `' ( { R }  +c  { S } ) `  k )  =  R )
2221fveq2d 5883 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  R )
)
2321fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  R )
)
24 xpsds.x . . . . . . . . . 10  |-  X  =  ( Base `  R
)
2523, 24syl6eqr 2523 . . . . . . . . 9  |-  ( (
ph  /\  k  =  (/) )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  X )
2625sqxpeqd 4865 . . . . . . . 8  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( X  X.  X ) )
2722, 26reseq12d 5112 . . . . . . 7  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  R )  |`  ( X  X.  X ) ) )
28 xpsds.m . . . . . . 7  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
2927, 28syl6eqr 2523 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  M )
3025fveq2d 5883 . . . . . 6  |-  ( (
ph  /\  k  =  (/) )  ->  ( *Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( *Met `  X
) )
3116, 29, 303eltr4d 2564 . . . . 5  |-  ( (
ph  /\  k  =  (/) )  ->  ( ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  e.  ( *Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) ) )
32 xpsds.4 . . . . . . 7  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
3332adantr 472 . . . . . 6  |-  ( (
ph  /\  k  =  1o )  ->  N  e.  ( *Met `  Y ) )
34 fveq2 5879 . . . . . . . . . 10  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
35 xpsds.2 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  W )
36 xpsc1 15545 . . . . . . . . . . 11  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
3735, 36syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
3834, 37sylan9eqr 2527 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  S )
3938fveq2d 5883 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  S )
)
4038fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
Base `  S )
)
41 xpsds.y . . . . . . . . . 10  |-  Y  =  ( Base `  S
)
4240, 41syl6eqr 2523 . . . . . . . . 9  |-  ( (
ph  /\  k  =  1o )  ->  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  =  Y )
4342sqxpeqd 4865 . . . . . . . 8  |-  ( (
ph  /\  k  =  1o )  ->  ( (
Base `  ( `' ( { R }  +c  { S } ) `  k ) )  X.  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) )  =  ( Y  X.  Y ) )
4439, 43reseq12d 5112 . . . . . . 7  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  ( ( dist `  S )  |`  ( Y  X.  Y ) ) )
45 xpsds.n . . . . . . 7  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
4644, 45syl6eqr 2523 . . . . . 6  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  =  N )
4742fveq2d 5883 . . . . . 6  |-  ( (
ph  /\  k  =  1o )  ->  ( *Met `  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) )  =  ( *Met `  Y ) )
4833, 46, 473eltr4d 2564 . . . . 5  |-  ( (
ph  /\  k  =  1o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  e.  ( *Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) ) )
4931, 48jaodan 802 . . . 4  |-  ( (
ph  /\  ( k  =  (/)  \/  k  =  1o ) )  -> 
( ( dist `  ( `' ( { R }  +c  { S }
) `  k )
)  |`  ( ( Base `  ( `' ( { R }  +c  { S } ) `  k
) )  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k ) ) ) )  e.  ( *Met `  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )
5014, 49sylan2 482 . . 3  |-  ( (
ph  /\  k  e.  2o )  ->  ( (
dist `  ( `' ( { R }  +c  { S } ) `  k ) )  |`  ( ( Base `  ( `' ( { R }  +c  { S }
) `  k )
)  X.  ( Base `  ( `' ( { R }  +c  { S } ) `  k
) ) ) )  e.  ( *Met `  ( Base `  ( `' ( { R }  +c  { S }
) `  k )
) ) )
511, 2, 3, 4, 5, 7, 9, 11, 50prdsxmet 21462 . 2  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )  e.  ( *Met `  ( Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) ) )
52 xpscfn 15543 . . . . . 6  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
5318, 35, 52syl2anc 673 . . . . 5  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
54 dffn5 5924 . . . . 5  |-  ( `' ( { R }  +c  { S } )  Fn  2o  <->  `' ( { R }  +c  { S } )  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) )
5553, 54sylib 201 . . . 4  |-  ( ph  ->  `' ( { R }  +c  { S }
)  =  ( k  e.  2o  |->  ( `' ( { R }  +c  { S } ) `
 k ) ) )
5655oveq2d 6324 . . 3  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) )
5756fveq2d 5883 . 2  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
dist `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
58 xpsds.t . . . . 5  |-  T  =  ( R  X.s  S )
59 eqid 2471 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
60 eqid 2471 . . . . 5  |-  (Scalar `  R )  =  (Scalar `  R )
61 eqid 2471 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
6258, 24, 41, 18, 35, 59, 60, 61xpslem 15557 . . . 4  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
6356fveq2d 5883 . . . 4  |-  ( ph  ->  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  =  (
Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
6462, 63eqtrd 2505 . . 3  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) )
6564fveq2d 5883 . 2  |-  ( ph  ->  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  =  ( *Met `  ( Base `  ( (Scalar `  R ) X_s ( k  e.  2o  |->  ( `' ( { R }  +c  { S }
) `  k )
) ) ) ) )
6651, 57, 653eltr4d 2564 1  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   {csn 3959   {cpr 3961    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   ran crn 4840    |` cres 4841   Oncon0 5430    Fn wfn 5584   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1oc1o 7193   2oc2o 7194    +c ccda 8615   Basecbs 15199  Scalarcsca 15271   distcds 15277   X_scprds 15422    X.s cxps 15483   *Metcxmt 19032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-prds 15424  df-xmet 19040
This theorem is referenced by:  xpsxmet  21473  xpsdsval  21474
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