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Theorem mrsub0 30166
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 3738 . . 3  |-  ( F  e.  ran  S  ->  -.  ran  S  =  (/) )
2 mrsubccat.s . . . . . 6  |-  S  =  (mRSubst `  T )
3 fvprc 5864 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
42, 3syl5eq 2499 . . . . 5  |-  ( -.  T  e.  _V  ->  S  =  (/) )
54rneqd 5065 . . . 4  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
6 rn0 5089 . . . 4  |-  ran  (/)  =  (/)
75, 6syl6eq 2503 . . 3  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
81, 7nsyl2 131 . 2  |-  ( F  e.  ran  S  ->  T  e.  _V )
9 eqid 2453 . . . . 5  |-  (mVR `  T )  =  (mVR
`  T )
10 eqid 2453 . . . . 5  |-  (mREx `  T )  =  (mREx `  T )
119, 10, 2mrsubff 30162 . . . 4  |-  ( T  e.  _V  ->  S : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
) )
12 ffun 5736 . . . 4  |-  ( S : ( (mREx `  T )  ^pm  (mVR `  T ) ) --> ( (mREx `  T )  ^m  (mREx `  T )
)  ->  Fun  S )
138, 11, 123syl 18 . . 3  |-  ( F  e.  ran  S  ->  Fun  S )
149, 10, 2mrsubrn 30163 . . . . 5  |-  ran  S  =  ( S "
( (mREx `  T
)  ^m  (mVR `  T
) ) )
1514eleq2i 2523 . . . 4  |-  ( F  e.  ran  S  <->  F  e.  ( S " ( (mREx `  T )  ^m  (mVR `  T ) ) ) )
1615biimpi 198 . . 3  |-  ( F  e.  ran  S  ->  F  e.  ( S " ( (mREx `  T
)  ^m  (mVR `  T
) ) ) )
17 fvelima 5922 . . 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 667 . 2  |-  ( F  e.  ran  S  ->  E. f  e.  (
(mREx `  T )  ^m  (mVR `  T )
) ( S `  f )  =  F )
19 elmapi 7498 . . . . . . 7  |-  ( f  e.  ( (mREx `  T )  ^m  (mVR `  T ) )  -> 
f : (mVR `  T ) --> (mREx `  T ) )
2019adantl 468 . . . . . 6  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  f : (mVR
`  T ) --> (mREx `  T ) )
21 ssid 3453 . . . . . . 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 12698 . . . . . . 7  |-  (/)  e. Word  (
(mCN `  T )  u.  (mVR `  T )
)
24 eqid 2453 . . . . . . . . 9  |-  (mCN `  T )  =  (mCN
`  T )
2524, 9, 10mrexval 30151 . . . . . . . 8  |-  ( T  e.  _V  ->  (mREx `  T )  = Word  (
(mCN `  T )  u.  (mVR `  T )
) )
2625adantr 467 . . . . . . 7  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  (mREx `  T
)  = Word  ( (mCN `  T )  u.  (mVR `  T ) ) )
2723, 26syl5eleqr 2538 . . . . . 6  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  (/)  e.  (mREx `  T ) )
28 eqid 2453 . . . . . . 7  |-  (freeMnd `  (
(mCN `  T )  u.  (mVR `  T )
) )  =  (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) )
2924, 9, 10, 2, 28mrsubval 30159 . . . . . 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 1269 . . . . 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 5352 . . . . . . 7  |-  ( ( v  e.  ( (mCN
`  T )  u.  (mVR `  T )
)  |->  if ( v  e.  (mVR `  T
) ,  ( f `
 v ) , 
<" v "> ) )  o.  (/) )  =  (/)
3231oveq2i 6306 . . . . . 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 16656 . . . . . . 7  |-  (/)  =  ( 0g `  (freeMnd `  (
(mCN `  T )  u.  (mVR `  T )
) ) )
3433gsum0 16533 . . . . . 6  |-  ( (freeMnd `  ( (mCN `  T
)  u.  (mVR `  T ) ) ) 
gsumg  (/) )  =  (/)
3532, 34eqtri 2475 . . . . 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 2503 . . . 4  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  ( ( S `
 f ) `  (/) )  =  (/) )
37 fveq1 5869 . . . . 5  |-  ( ( S `  f )  =  F  ->  (
( S `  f
) `  (/) )  =  ( F `  (/) ) )
3837eqeq1d 2455 . . . 4  |-  ( ( S `  f )  =  F  ->  (
( ( S `  f ) `  (/) )  =  (/) 
<->  ( F `  (/) )  =  (/) ) )
3936, 38syl5ibcom 224 . . 3  |-  ( ( T  e.  _V  /\  f  e.  ( (mREx `  T )  ^m  (mVR `  T ) ) )  ->  ( ( S `
 f )  =  F  ->  ( F `  (/) )  =  (/) ) )
4039rexlimdva 2881 . 2  |-  ( T  e.  _V  ->  ( E. f  e.  (
(mREx `  T )  ^m  (mVR `  T )
) ( S `  f )  =  F  ->  ( F `  (/) )  =  (/) ) )
418, 18, 40sylc 62 1  |-  ( F  e.  ran  S  -> 
( F `  (/) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   E.wrex 2740   _Vcvv 3047    u. cun 3404    C_ wss 3406   (/)c0 3733   ifcif 3883    |-> cmpt 4464   ran crn 4838   "cima 4840    o. ccom 4841   Fun wfun 5579   -->wf 5581   ` cfv 5585  (class class class)co 6295    ^m cmap 7477    ^pm cpm 7478  Word cword 12663   <"cs1 12666    gsumg cgsu 15351  freeMndcfrmd 16643  mCNcmcn 30110  mVRcmvar 30111  mRExcmrex 30116  mRSubstcmrsub 30120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12221  df-hash 12523  df-word 12671  df-concat 12673  df-s1 12674  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-0g 15352  df-gsum 15353  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-frmd 16645  df-mrex 30136  df-mrsub 30140
This theorem is referenced by:  mrsubvrs  30172
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