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Theorem mrsub0 31413
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 3920 . . 3 |- ( F e. ran S -> -. ran S = (/) )
2 mrsubccat.s . . . . . 6 |- S = (mRSubst ` T )
3 fvprc 6185 . . . . . 6 |- ( -. T e. _V -> (mRSubst ` T ) = (/) )
42, 3syl5eq 2668 . . . . 5 |- ( -. T e. _V -> S = (/) )
54rneqd 5353 . . . 4 |- ( -. T e. _V -> ran S = ran (/) )
6 rn0 5377 . . . 4 |- ran (/) = (/)
75, 6syl6eq 2672 . . 3 |- ( -. T e. _V -> ran S = (/) )
81, 7nsyl2 142 . 2 |- ( F e. ran S -> T e. _V )
9 eqid 2622 . . . . 5 |- (mVR ` T ) = (mVR ` T )
10 eqid 2622 . . . . 5 |- (mREx ` T ) = (mREx ` T )
119, 10, 2mrsubff 31409 . . . 4 |- ( T e. _V -> S : ( (mREx ` T ) ^pm (mVR ` T ) ) --> ( (mREx ` T ) ^m (mREx ` T ) ) )
12 ffun 6048 . . . 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 31410 . . . . 5 |- ran S = ( S " ( (mREx ` T ) ^m (mVR ` T ) ) )
1514eleq2i 2693 . . . 4 |- ( F e. ran S <-> F e. ( S " ( (mREx ` T ) ^m (mVR ` T ) ) ) )
1615biimpi 206 . . 3 |- ( F e. ran S -> F e. ( S " ( (mREx ` T ) ^m (mVR ` T ) ) ) )
17 fvelima 6248 . . 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 693 . 2 |- ( F e. ran S -> E. f e. ( (mREx ` T ) ^m (mVR ` T ) ) ( S ` f ) = F )
19 elmapi 7879 . . . . . . 7 |- ( f e. ( (mREx ` T ) ^m (mVR ` T ) ) -> f : (mVR ` T ) --> (mREx ` T ) )
2019adantl 482 . . . . . 6 |- ( ( T e. _V /\ f e. ( (mREx ` T ) ^m (mVR ` T ) ) ) -> f : (mVR ` T ) --> (mREx ` T ) )
21 ssid 3624 . . . . . . 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 13330 . . . . . . 7 |- (/) e. Word ( (mCN ` T ) u. (mVR ` T ) )
24 eqid 2622 . . . . . . . . 9 |- (mCN ` T ) = (mCN ` T )
2524, 9, 10mrexval 31398 . . . . . . . 8 |- ( T e. _V -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
2625adantr 481 . . . . . . 7 |- ( ( T e. _V /\ f e. ( (mREx ` T ) ^m (mVR ` T ) ) ) -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
2723, 26syl5eleqr 2708 . . . . . 6 |- ( ( T e. _V /\ f e. ( (mREx ` T ) ^m (mVR ` T ) ) ) -> (/) e. (mREx ` T ) )
28 eqid 2622 . . . . . . 7 |- (freeMnd ` ( (mCN ` T ) u. (mVR ` T ) ) ) = (freeMnd ` ( (mCN ` T ) u. (mVR ` T ) ) )
2924, 9, 10, 2, 28mrsubval 31406 . . . . . 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 1326 . . . . 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 5649 . . . . . . 7 |- ( ( v e. ( (mCN ` T ) u. (mVR ` T ) ) |-> if ( v e. (mVR ` T ) , ( f ` v ) , <" v "> ) ) o. (/) ) = (/)
3231oveq2i 6661 . . . . . 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 17397 . . . . . . 7 |- (/) = ( 0g ` (freeMnd ` ( (mCN ` T ) u. (mVR ` T ) ) ) )
3433gsum0 17278 . . . . . 6 |- ( (freeMnd ` ( (mCN ` T ) u. (mVR ` T ) ) ) gsumg (/) ) = (/)
3532, 34eqtri 2644 . . . . 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 2672 . . . 4 |- ( ( T e. _V /\ f e. ( (mREx ` T ) ^m (mVR ` T ) ) ) -> ( ( S ` f ) ` (/) ) = (/) )
37 fveq1 6190 . . . . 5 |- ( ( S ` f ) = F -> ( ( S ` f ) ` (/) ) = ( F ` (/) ) )
3837eqeq1d 2624 . . . 4 |- ( ( S ` f ) = F -> ( ( ( S ` f ) ` (/) ) = (/) <-> ( F ` (/) ) = (/) ) )
3936, 38syl5ibcom 235 . . 3 |- ( ( T e. _V /\ f e. ( (mREx ` T ) ^m (mVR ` T ) ) ) -> ( ( S ` f ) = F -> ( F ` (/) ) = (/) ) )
4039rexlimdva 3031 . 2 |- ( T e. _V -> ( E. f e. ( (mREx ` T ) ^m (mVR ` T ) ) ( S ` f ) = F -> ( F ` (/) ) = (/) ) )
418, 18, 40sylc 65 1 |- ( F e. ran S -> ( F ` (/) ) = (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   |-> cmpt 4729   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   ^pm cpm 7858  Word cword 13291   <"cs1 13294   gsumg cgsu 16101  freeMndcfrmd 17384  mCNcmcn 31357  mVRcmvar 31358  mRExcmrex 31363  mRSubstcmrsub 31367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-mrex 31383  df-mrsub 31387
This theorem is referenced by:  mrsubvrs  31419
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