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Theorem mrsubco 29145
Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mrsubco.s  |-  S  =  (mRSubst `  T )
Assertion
Ref Expression
mrsubco  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  ( F  o.  G )  e.  ran  S )

Proof of Theorem mrsubco
Dummy variables  c  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrsubco.s . . . . 5  |-  S  =  (mRSubst `  T )
2 eqid 2454 . . . . 5  |-  (mREx `  T )  =  (mREx `  T )
31, 2mrsubf 29141 . . . 4  |-  ( F  e.  ran  S  ->  F : (mREx `  T
) --> (mREx `  T
) )
43adantr 463 . . 3  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  F :
(mREx `  T ) --> (mREx `  T ) )
51, 2mrsubf 29141 . . . 4  |-  ( G  e.  ran  S  ->  G : (mREx `  T
) --> (mREx `  T
) )
65adantl 464 . . 3  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  G :
(mREx `  T ) --> (mREx `  T ) )
7 fco 5723 . . 3  |-  ( ( F : (mREx `  T ) --> (mREx `  T )  /\  G : (mREx `  T ) --> (mREx `  T ) )  ->  ( F  o.  G ) : (mREx `  T ) --> (mREx `  T ) )
84, 6, 7syl2anc 659 . 2  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  ( F  o.  G ) : (mREx `  T ) --> (mREx `  T ) )
96adantr 463 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  ->  G : (mREx `  T
) --> (mREx `  T
) )
10 eldifi 3612 . . . . . . . . 9  |-  ( c  e.  ( (mCN `  T )  \  (mVR `  T ) )  -> 
c  e.  (mCN `  T ) )
11 elun1 3657 . . . . . . . . 9  |-  ( c  e.  (mCN `  T
)  ->  c  e.  ( (mCN `  T )  u.  (mVR `  T )
) )
1210, 11syl 16 . . . . . . . 8  |-  ( c  e.  ( (mCN `  T )  \  (mVR `  T ) )  -> 
c  e.  ( (mCN
`  T )  u.  (mVR `  T )
) )
1312adantl 464 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
c  e.  ( (mCN
`  T )  u.  (mVR `  T )
) )
1413s1cld 12604 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  ->  <" c ">  e. Word  ( (mCN `  T
)  u.  (mVR `  T ) ) )
15 n0i 3788 . . . . . . . . . 10  |-  ( F  e.  ran  S  ->  -.  ran  S  =  (/) )
16 fvprc 5842 . . . . . . . . . . . . 13  |-  ( -.  T  e.  _V  ->  (mRSubst `  T )  =  (/) )
171, 16syl5eq 2507 . . . . . . . . . . . 12  |-  ( -.  T  e.  _V  ->  S  =  (/) )
1817rneqd 5219 . . . . . . . . . . 11  |-  ( -.  T  e.  _V  ->  ran 
S  =  ran  (/) )
19 rn0 5243 . . . . . . . . . . 11  |-  ran  (/)  =  (/)
2018, 19syl6eq 2511 . . . . . . . . . 10  |-  ( -.  T  e.  _V  ->  ran 
S  =  (/) )
2115, 20nsyl2 127 . . . . . . . . 9  |-  ( F  e.  ran  S  ->  T  e.  _V )
2221adantr 463 . . . . . . . 8  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  T  e.  _V )
2322adantr 463 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  ->  T  e.  _V )
24 eqid 2454 . . . . . . . 8  |-  (mCN `  T )  =  (mCN
`  T )
25 eqid 2454 . . . . . . . 8  |-  (mVR `  T )  =  (mVR
`  T )
2624, 25, 2mrexval 29125 . . . . . . 7  |-  ( T  e.  _V  ->  (mREx `  T )  = Word  (
(mCN `  T )  u.  (mVR `  T )
) )
2723, 26syl 16 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
(mREx `  T )  = Word  ( (mCN `  T
)  u.  (mVR `  T ) ) )
2814, 27eleqtrrd 2545 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  ->  <" c ">  e.  (mREx `  T )
)
29 fvco3 5925 . . . . 5  |-  ( ( G : (mREx `  T ) --> (mREx `  T )  /\  <" c ">  e.  (mREx `  T ) )  ->  ( ( F  o.  G ) `  <" c "> )  =  ( F `  ( G `  <" c "> )
) )
309, 28, 29syl2anc 659 . . . 4  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
( ( F  o.  G ) `  <" c "> )  =  ( F `  ( G `  <" c "> ) ) )
311, 2, 25, 24mrsubcn 29143 . . . . . 6  |-  ( ( G  e.  ran  S  /\  c  e.  (
(mCN `  T )  \  (mVR `  T )
) )  ->  ( G `  <" c "> )  =  <" c "> )
3231adantll 711 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
( G `  <" c "> )  =  <" c "> )
3332fveq2d 5852 . . . 4  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
( F `  ( G `  <" c "> ) )  =  ( F `  <" c "> )
)
341, 2, 25, 24mrsubcn 29143 . . . . 5  |-  ( ( F  e.  ran  S  /\  c  e.  (
(mCN `  T )  \  (mVR `  T )
) )  ->  ( F `  <" c "> )  =  <" c "> )
3534adantlr 712 . . . 4  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
( F `  <" c "> )  =  <" c "> )
3630, 33, 353eqtrd 2499 . . 3  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) )  -> 
( ( F  o.  G ) `  <" c "> )  =  <" c "> )
3736ralrimiva 2868 . 2  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  A. c  e.  ( (mCN `  T
)  \  (mVR `  T
) ) ( ( F  o.  G ) `
 <" c "> )  =  <" c "> )
381, 2mrsubccat 29142 . . . . . . . 8  |-  ( ( G  e.  ran  S  /\  x  e.  (mREx `  T )  /\  y  e.  (mREx `  T )
)  ->  ( G `  ( x ++  y ) )  =  ( ( G `  x ) ++  ( G `  y
) ) )
39383expb 1195 . . . . . . 7  |-  ( ( G  e.  ran  S  /\  ( x  e.  (mREx `  T )  /\  y  e.  (mREx `  T )
) )  ->  ( G `  ( x ++  y ) )  =  ( ( G `  x ) ++  ( G `  y ) ) )
4039adantll 711 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( G `  ( x ++  y ) )  =  ( ( G `  x ) ++  ( G `  y
) ) )
4140fveq2d 5852 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( F `  ( G `  (
x ++  y ) ) )  =  ( F `
 ( ( G `
 x ) ++  ( G `  y ) ) ) )
42 simpll 751 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  F  e.  ran  S )
436adantr 463 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  G :
(mREx `  T ) --> (mREx `  T ) )
44 simprl 754 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  x  e.  (mREx `  T ) )
4543, 44ffvelrnd 6008 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( G `  x )  e.  (mREx `  T ) )
46 simprr 755 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  y  e.  (mREx `  T ) )
4743, 46ffvelrnd 6008 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( G `  y )  e.  (mREx `  T ) )
481, 2mrsubccat 29142 . . . . . 6  |-  ( ( F  e.  ran  S  /\  ( G `  x
)  e.  (mREx `  T )  /\  ( G `  y )  e.  (mREx `  T )
)  ->  ( F `  ( ( G `  x ) ++  ( G `  y ) ) )  =  ( ( F `
 ( G `  x ) ) ++  ( F `  ( G `
 y ) ) ) )
4942, 45, 47, 48syl3anc 1226 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( F `  ( ( G `  x ) ++  ( G `  y ) ) )  =  ( ( F `
 ( G `  x ) ) ++  ( F `  ( G `
 y ) ) ) )
5041, 49eqtrd 2495 . . . 4  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( F `  ( G `  (
x ++  y ) ) )  =  ( ( F `  ( G `
 x ) ) ++  ( F `  ( G `  y )
) ) )
5122, 26syl 16 . . . . . . . . 9  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  (mREx `  T
)  = Word  ( (mCN `  T )  u.  (mVR `  T ) ) )
5251adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  (mREx `  T
)  = Word  ( (mCN `  T )  u.  (mVR `  T ) ) )
5344, 52eleqtrd 2544 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  x  e. Word  ( (mCN `  T )  u.  (mVR `  T )
) )
5446, 52eleqtrd 2544 . . . . . . 7  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  y  e. Word  ( (mCN `  T )  u.  (mVR `  T )
) )
55 ccatcl 12582 . . . . . . 7  |-  ( ( x  e. Word  ( (mCN
`  T )  u.  (mVR `  T )
)  /\  y  e. Word  ( (mCN `  T )  u.  (mVR `  T )
) )  ->  (
x ++  y )  e. Word 
( (mCN `  T
)  u.  (mVR `  T ) ) )
5653, 54, 55syl2anc 659 . . . . . 6  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( x ++  y )  e. Word  (
(mCN `  T )  u.  (mVR `  T )
) )
5756, 52eleqtrrd 2545 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( x ++  y )  e.  (mREx `  T ) )
58 fvco3 5925 . . . . 5  |-  ( ( G : (mREx `  T ) --> (mREx `  T )  /\  (
x ++  y )  e.  (mREx `  T )
)  ->  ( ( F  o.  G ) `  ( x ++  y ) )  =  ( F `
 ( G `  ( x ++  y )
) ) )
5943, 57, 58syl2anc 659 . . . 4  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( ( F  o.  G ) `  ( x ++  y ) )  =  ( F `
 ( G `  ( x ++  y )
) ) )
60 fvco3 5925 . . . . . 6  |-  ( ( G : (mREx `  T ) --> (mREx `  T )  /\  x  e.  (mREx `  T )
)  ->  ( ( F  o.  G ) `  x )  =  ( F `  ( G `
 x ) ) )
6143, 44, 60syl2anc 659 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( ( F  o.  G ) `  x )  =  ( F `  ( G `
 x ) ) )
62 fvco3 5925 . . . . . 6  |-  ( ( G : (mREx `  T ) --> (mREx `  T )  /\  y  e.  (mREx `  T )
)  ->  ( ( F  o.  G ) `  y )  =  ( F `  ( G `
 y ) ) )
6343, 46, 62syl2anc 659 . . . . 5  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( ( F  o.  G ) `  y )  =  ( F `  ( G `
 y ) ) )
6461, 63oveq12d 6288 . . . 4  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( (
( F  o.  G
) `  x ) ++  ( ( F  o.  G ) `  y
) )  =  ( ( F `  ( G `  x )
) ++  ( F `  ( G `  y ) ) ) )
6550, 59, 643eqtr4d 2505 . . 3  |-  ( ( ( F  e.  ran  S  /\  G  e.  ran  S )  /\  ( x  e.  (mREx `  T
)  /\  y  e.  (mREx `  T ) ) )  ->  ( ( F  o.  G ) `  ( x ++  y ) )  =  ( ( ( F  o.  G
) `  x ) ++  ( ( F  o.  G ) `  y
) ) )
6665ralrimivva 2875 . 2  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  A. x  e.  (mREx `  T ) A. y  e.  (mREx `  T ) ( ( F  o.  G ) `
 ( x ++  y
) )  =  ( ( ( F  o.  G ) `  x
) ++  ( ( F  o.  G ) `  y ) ) )
671, 2, 25, 24elmrsubrn 29144 . . 3  |-  ( T  e.  _V  ->  (
( F  o.  G
)  e.  ran  S  <->  ( ( F  o.  G
) : (mREx `  T ) --> (mREx `  T )  /\  A. c  e.  ( (mCN `  T )  \  (mVR `  T ) ) ( ( F  o.  G
) `  <" c "> )  =  <" c ">  /\  A. x  e.  (mREx `  T
) A. y  e.  (mREx `  T )
( ( F  o.  G ) `  (
x ++  y ) )  =  ( ( ( F  o.  G ) `
 x ) ++  ( ( F  o.  G
) `  y )
) ) ) )
6822, 67syl 16 . 2  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  ( ( F  o.  G )  e.  ran  S  <->  ( ( F  o.  G ) : (mREx `  T ) --> (mREx `  T )  /\  A. c  e.  ( (mCN
`  T )  \ 
(mVR `  T )
) ( ( F  o.  G ) `  <" c "> )  =  <" c ">  /\  A. x  e.  (mREx `  T ) A. y  e.  (mREx `  T ) ( ( F  o.  G ) `
 ( x ++  y
) )  =  ( ( ( F  o.  G ) `  x
) ++  ( ( F  o.  G ) `  y ) ) ) ) )
698, 37, 66, 68mpbir3and 1177 1  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  ( F  o.  G )  e.  ran  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    \ cdif 3458    u. cun 3459   (/)c0 3783   ran crn 4989    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270  Word cword 12518   ++ cconcat 12520   <"cs1 12521  mCNcmcn 29084  mVRcmvar 29085  mRExcmrex 29090  mRSubstcmrsub 29094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-frmd 16216  df-vrmd 16217  df-mrex 29110  df-mrsub 29114
This theorem is referenced by:  msubco  29155
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