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Theorem mrsub0 29160
Description: The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mrsubccat.s  |-  S  =  (mRSubst `  T )
Assertion
Ref Expression
mrsub0  |-  ( F  e.  ran  S  -> 
( F `  (/) )  =  (/) )

Proof of Theorem mrsub0
Dummy variables  f 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3798 . . 3  |-  ( F  e.  ran  S  ->  -.  ran  S  =  (/) )
2 mrsubccat.s . . . . . 6  |-  S  =  (mRSubst `  T )
3 fvprc 5866 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
42, 3syl5eq 2510 . . . . 5  |-  ( -.  T  e.  _V  ->  S  =  (/) )
54rneqd 5240 . . . 4  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
6 rn0 5264 . . . 4  |-  ran  (/)  =  (/)
75, 6syl6eq 2514 . . 3  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
81, 7nsyl2 127 . 2  |-  ( F  e.  ran  S  ->  T  e.  _V )
9 eqid 2457 . . . . 5  |-  (mVR `  T )  =  (mVR
`  T )
10 eqid 2457 . . . . 5  |-  (mREx `  T )  =  (mREx `  T )
119, 10, 2mrsubff 29156 . . . 4  |-  ( T  e.  _V  ->  S : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
) )
12 ffun 5739 . . . 4  |-  ( S : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
)  ->  Fun  S )
138, 11, 123syl 20 . . 3  |-  ( F  e.  ran  S  ->  Fun  S )
149, 10, 2mrsubrn 29157 . . . . 5  |-  ran  S  =  ( S "
( (mREx `  T
)  ^m  (mVR `  T
) ) )
1514eleq2i 2535 . . . 4  |-  ( F  e.  ran  S  <->  F  e.  ( S " ( (mREx `  T )  ^m  (mVR `  T ) ) ) )
1615biimpi 194 . . 3  |-  ( F  e.  ran  S  ->  F  e.  ( S " ( (mREx `  T
)  ^m  (mVR `  T
) ) ) )
17 fvelima 5925 . . 3  |-  ( ( Fun  S  /\  F  e.  ( S " (
(mREx `  T )  ^m  (mVR `  T )
) ) )  ->  E. f  e.  (
(mREx `  T )  ^m  (mVR `  T )
) ( S `  f )  =  F )
1813, 16, 17syl2anc 661 . 2  |-  ( F  e.  ran  S  ->  E. f  e.  (
(mREx `  T )  ^m  (mVR `  T )
) ( S `  f )  =  F )
19 elmapi 7459 . . . . . . 7  |-  ( f  e.  ( (mREx `  T )  ^m  (mVR `  T ) )  -> 
f : (mVR `  T ) --> (mREx `  T ) )
2019adantl 466 . . . . . 6  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  f : (mVR
`  T ) --> (mREx `  T ) )
21 ssid 3518 . . . . . . 7  |-  (mVR `  T )  C_  (mVR `  T )
2221a1i 11 . . . . . 6  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  (mVR `  T
)  C_  (mVR `  T
) )
23 wrd0 12573 . . . . . . 7  |-  (/)  e. Word  (
(mCN `  T )  u.  (mVR `  T )
)
24 eqid 2457 . . . . . . . . 9  |-  (mCN `  T )  =  (mCN
`  T )
2524, 9, 10mrexval 29145 . . . . . . . 8  |-  ( T  e.  _V  ->  (mREx `  T )  = Word  (
(mCN `  T )  u.  (mVR `  T )
) )
2625adantr 465 . . . . . . 7  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  (mREx `  T
)  = Word  ( (mCN `  T )  u.  (mVR `  T ) ) )
2723, 26syl5eleqr 2552 . . . . . 6  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  (/)  e.  (mREx `  T ) )
28 eqid 2457 . . . . . . 7  |-  (freeMnd `  (
(mCN `  T )  u.  (mVR `  T )
) )  =  (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) )
2924, 9, 10, 2, 28mrsubval 29153 . . . . . 6  |-  ( ( f : (mVR `  T ) --> (mREx `  T )  /\  (mVR `  T )  C_  (mVR `  T )  /\  (/)  e.  (mREx `  T ) )  -> 
( ( S `  f ) `  (/) )  =  ( (freeMnd `  (
(mCN `  T )  u.  (mVR `  T )
) )  gsumg  ( ( v  e.  ( (mCN `  T
)  u.  (mVR `  T ) )  |->  if ( v  e.  (mVR
`  T ) ,  ( f `  v
) ,  <" v "> ) )  o.  (/) ) ) )
3020, 22, 27, 29syl3anc 1228 . . . . 5  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  ( ( S `
 f ) `  (/) )  =  ( (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  T
)  u.  (mVR `  T ) )  |->  if ( v  e.  (mVR
`  T ) ,  ( f `  v
) ,  <" v "> ) )  o.  (/) ) ) )
31 co02 5527 . . . . . . 7  |-  ( ( v  e.  ( (mCN
`  T )  u.  (mVR `  T )
)  |->  if ( v  e.  (mVR `  T
) ,  ( f `
 v ) , 
<" v "> ) )  o.  (/) )  =  (/)
3231oveq2i 6307 . . . . . 6  |-  ( (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  T
)  u.  (mVR `  T ) )  |->  if ( v  e.  (mVR
`  T ) ,  ( f `  v
) ,  <" v "> ) )  o.  (/) ) )  =  ( (freeMnd `  ( (mCN `  T )  u.  (mVR `  T ) ) ) 
gsumg  (/) )
3328frmd0 16246 . . . . . . 7  |-  (/)  =  ( 0g `  (freeMnd `  (
(mCN `  T )  u.  (mVR `  T )
) ) )
3433gsum0 16123 . . . . . 6  |-  ( (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) ) 
gsumg  (/) )  =  (/)
3532, 34eqtri 2486 . . . . 5  |-  ( (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  T
)  u.  (mVR `  T ) )  |->  if ( v  e.  (mVR
`  T ) ,  ( f `  v
) ,  <" v "> ) )  o.  (/) ) )  =  (/)
3630, 35syl6eq 2514 . . . 4  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  ( ( S `
 f ) `  (/) )  =  (/) )
37 fveq1 5871 . . . . 5  |-  ( ( S `  f )  =  F  ->  (
( S `  f
) `  (/) )  =  ( F `  (/) ) )
3837eqeq1d 2459 . . . 4  |-  ( ( S `  f )  =  F  ->  (
( ( S `  f ) `  (/) )  =  (/) 
<->  ( F `  (/) )  =  (/) ) )
3936, 38syl5ibcom 220 . . 3  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  ( ( S `
 f )  =  F  ->  ( F `  (/) )  =  (/) ) )
4039rexlimdva 2949 . 2  |-  ( T  e.  _V  ->  ( E. f  e.  (
(mREx `  T )  ^m  (mVR `  T )
) ( S `  f )  =  F  ->  ( F `  (/) )  =  (/) ) )
418, 18, 40sylc 60 1  |-  ( F  e.  ran  S  -> 
( F `  (/) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    u. cun 3469    C_ wss 3471   (/)c0 3793   ifcif 3944    |-> cmpt 4515   ran crn 5009   "cima 5011    o. ccom 5012   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438    ^pm cpm 7439  Word cword 12538   <"cs1 12541    gsumg cgsu 14949  freeMndcfrmd 16233  mCNcmcn 29104  mVRcmvar 29105  mRExcmrex 29110  mRSubstcmrsub 29114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-ress 14742  df-plusg 14816  df-0g 14950  df-gsum 14951  df-mgm 16090  df-sgrp 16129  df-mnd 16139  df-submnd 16185  df-frmd 16235  df-mrex 29130  df-mrsub 29134
This theorem is referenced by:  mrsubvrs  29166
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