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Theorem frlmup4 26922
Description: Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypotheses
Ref Expression
frlmup4.r  |-  R  =  (Scalar `  T )
frlmup4.f  |-  F  =  ( R freeLMod  I )
frlmup4.u  |-  U  =  ( R unitVec  I )
frlmup4.c  |-  C  =  ( Base `  T
)
Assertion
Ref Expression
frlmup4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E! m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
Distinct variable groups:    A, m    m, F    T, m    U, m
Allowed substitution hints:    C( m)    R( m)    I( m)    X( m)

Proof of Theorem frlmup4
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmup4.f . . . 4  |-  F  =  ( R freeLMod  I )
2 eqid 2387 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
3 frlmup4.c . . . 4  |-  C  =  ( Base `  T
)
4 eqid 2387 . . . 4  |-  ( .s
`  T )  =  ( .s `  T
)
5 eqid 2387 . . . 4  |-  ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  =  ( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )
6 simp1 957 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  T  e.  LMod )
7 simp2 958 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  I  e.  X )
8 frlmup4.r . . . . 5  |-  R  =  (Scalar `  T )
98a1i 11 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  R  =  (Scalar `  T
) )
10 simp3 959 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  A : I --> C )
111, 2, 3, 4, 5, 6, 7, 9, 10frlmup1 26919 . . 3  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  e.  ( F LMHom 
T ) )
12 ovex 6045 . . . . . . 7  |-  ( T 
gsumg  ( x  o F
( .s `  T
) A ) )  e.  _V
1312, 5fnmpti 5513 . . . . . 6  |-  ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  Fn  ( Base `  F
)
1413a1i 11 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  Fn  ( Base `  F ) )
158lmodrng 15885 . . . . . . . 8  |-  ( T  e.  LMod  ->  R  e. 
Ring )
16153ad2ant1 978 . . . . . . 7  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  R  e.  Ring )
17 frlmup4.u . . . . . . . 8  |-  U  =  ( R unitVec  I )
1817, 1, 2uvcff 26909 . . . . . . 7  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  U : I --> ( Base `  F ) )
1916, 7, 18syl2anc 643 . . . . . 6  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  U : I --> ( Base `  F ) )
20 ffn 5531 . . . . . 6  |-  ( U : I --> ( Base `  F )  ->  U  Fn  I )
2119, 20syl 16 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  U  Fn  I )
22 frn 5537 . . . . . 6  |-  ( U : I --> ( Base `  F )  ->  ran  U 
C_  ( Base `  F
) )
2319, 22syl 16 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  ran  U  C_  ( Base `  F ) )
24 fnco 5493 . . . . 5  |-  ( ( ( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  Fn  ( Base `  F )  /\  U  Fn  I  /\  ran  U  C_  ( Base `  F
) )  ->  (
( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  o.  U )  Fn  I )
2514, 21, 23, 24syl3anc 1184 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  o.  U )  Fn  I
)
26 ffn 5531 . . . . 5  |-  ( A : I --> C  ->  A  Fn  I )
27263ad2ant3 980 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  A  Fn  I )
2819adantr 452 . . . . . . 7  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  U :
I --> ( Base `  F
) )
2928, 20syl 16 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  U  Fn  I )
30 simpr 448 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  y  e.  I )
31 fvco2 5737 . . . . . 6  |-  ( ( U  Fn  I  /\  y  e.  I )  ->  ( ( ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  o.  U ) `  y
)  =  ( ( x  e.  ( Base `  F )  |->  ( T 
gsumg  ( x  o F
( .s `  T
) A ) ) ) `  ( U `
 y ) ) )
3229, 30, 31syl2anc 643 . . . . 5  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  ( (
( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  o.  U ) `
 y )  =  ( ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) ) `  ( U `  y ) ) )
33 simpl1 960 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  T  e.  LMod )
34 simpl2 961 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  I  e.  X )
358a1i 11 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  R  =  (Scalar `  T ) )
36 simpl3 962 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  A :
I --> C )
371, 2, 3, 4, 5, 33, 34, 35, 36, 30, 17frlmup2 26920 . . . . 5  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  ( (
x  e.  ( Base `  F )  |->  ( T 
gsumg  ( x  o F
( .s `  T
) A ) ) ) `  ( U `
 y ) )  =  ( A `  y ) )
3832, 37eqtrd 2419 . . . 4  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  ( (
( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  o.  U ) `
 y )  =  ( A `  y
) )
3925, 27, 38eqfnfvd 5769 . . 3  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  o.  U )  =  A )
40 coeq1 4970 . . . . 5  |-  ( m  =  ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  -> 
( m  o.  U
)  =  ( ( x  e.  ( Base `  F )  |->  ( T 
gsumg  ( x  o F
( .s `  T
) A ) ) )  o.  U ) )
4140eqeq1d 2395 . . . 4  |-  ( m  =  ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  -> 
( ( m  o.  U )  =  A  <-> 
( ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  o.  U )  =  A ) )
4241rspcev 2995 . . 3  |-  ( ( ( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  e.  ( F LMHom 
T )  /\  (
( x  e.  (
Base `  F )  |->  ( T  gsumg  ( x  o F ( .s `  T
) A ) ) )  o.  U )  =  A )  ->  E. m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
4311, 39, 42syl2anc 643 . 2  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E. m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
44 ffun 5533 . . . . 5  |-  ( U : I --> ( Base `  F )  ->  Fun  U )
4519, 44syl 16 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  Fun  U )
46 funcoeqres 5646 . . . . . 6  |-  ( ( Fun  U  /\  (
m  o.  U )  =  A )  -> 
( m  |`  ran  U
)  =  ( A  o.  `' U ) )
4746ex 424 . . . . 5  |-  ( Fun 
U  ->  ( (
m  o.  U )  =  A  ->  (
m  |`  ran  U )  =  ( A  o.  `' U ) ) )
4847ralrimivw 2733 . . . 4  |-  ( Fun 
U  ->  A. m  e.  ( F LMHom  T ) ( ( m  o.  U )  =  A  ->  ( m  |`  ran  U )  =  ( A  o.  `' U
) ) )
4945, 48syl 16 . . 3  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  A. m  e.  ( F LMHom  T ) ( ( m  o.  U )  =  A  ->  (
m  |`  ran  U )  =  ( A  o.  `' U ) ) )
50 eqid 2387 . . . . . . 7  |-  (LBasis `  F )  =  (LBasis `  F )
511, 17, 50frlmlbs 26918 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  ran  U  e.  (LBasis `  F
) )
5216, 7, 51syl2anc 643 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  ran  U  e.  (LBasis `  F ) )
53 eqid 2387 . . . . . 6  |-  ( LSpan `  F )  =  (
LSpan `  F )
542, 50, 53lbssp 16078 . . . . 5  |-  ( ran 
U  e.  (LBasis `  F )  ->  (
( LSpan `  F ) `  ran  U )  =  ( Base `  F
) )
5552, 54syl 16 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( ( LSpan `  F
) `  ran  U )  =  ( Base `  F
) )
562, 53lspextmo 16059 . . . 4  |-  ( ( ran  U  C_  ( Base `  F )  /\  ( ( LSpan `  F
) `  ran  U )  =  ( Base `  F
) )  ->  E* m  e.  ( F LMHom  T ) ( m  |`  ran  U )  =  ( A  o.  `' U
) )
5723, 55, 56syl2anc 643 . . 3  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E* m  e.  ( F LMHom  T ) ( m  |`  ran  U )  =  ( A  o.  `' U ) )
58 rmoim 3076 . . 3  |-  ( A. m  e.  ( F LMHom  T ) ( ( m  o.  U )  =  A  ->  ( m  |` 
ran  U )  =  ( A  o.  `' U ) )  -> 
( E* m  e.  ( F LMHom  T ) ( m  |`  ran  U
)  =  ( A  o.  `' U )  ->  E* m  e.  ( F LMHom  T ) ( m  o.  U
)  =  A ) )
5949, 57, 58sylc 58 . 2  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E* m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
60 reu5 2864 . 2  |-  ( E! m  e.  ( F LMHom 
T ) ( m  o.  U )  =  A  <->  ( E. m  e.  ( F LMHom  T ) ( m  o.  U
)  =  A  /\  E* m  e.  ( F LMHom  T ) ( m  o.  U )  =  A ) )
6143, 59, 60sylanbrc 646 1  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E! m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   E!wreu 2651   E*wrmo 2652    C_ wss 3263    e. cmpt 4207   `'ccnv 4817   ran crn 4819    |` cres 4820    o. ccom 4822   Fun wfun 5388    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Fcof 6242   Basecbs 13396  Scalarcsca 13459   .scvsca 13460    gsumg cgsu 13651   Ringcrg 15587   LModclmod 15877   LSpanclspn 15974   LMHom clmhm 16022  LBasisclbs 16073   freeLMod cfrlm 26881   unitVec cuvc 26882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-hom 13480  df-cco 13481  df-prds 13598  df-pws 13600  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-cntz 15043  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-subrg 15793  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lmhm 16025  df-lbs 16074  df-sra 16171  df-rgmod 16172  df-nzr 16256  df-dsmm 26867  df-frlm 26883  df-uvc 26884
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