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Theorem frlmup4 27356
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 2296 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
3 frlmup4.c . . . 4  |-  C  =  ( Base `  T
)
4 eqid 2296 . . . 4  |-  ( .s
`  T )  =  ( .s `  T
)
5 eqid 2296 . . . 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 955 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  T  e.  LMod )
7 simp2 956 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  I  e.  X )
8 frlmup4.r . . . . 5  |-  R  =  (Scalar `  T )
98a1i 10 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  R  =  (Scalar `  T
) )
10 simp3 957 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  A : I --> C )
111, 2, 3, 4, 5, 6, 7, 9, 10frlmup1 27353 . . 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 5899 . . . . . . 7  |-  ( T 
gsumg  ( x  o F
( .s `  T
) A ) )  e.  _V
1312, 5fnmpti 5388 . . . . . 6  |-  ( x  e.  ( Base `  F
)  |->  ( T  gsumg  ( x  o F ( .s
`  T ) A ) ) )  Fn  ( Base `  F
)
1413a1i 10 . . . . 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 15651 . . . . . . . 8  |-  ( T  e.  LMod  ->  R  e. 
Ring )
16153ad2ant1 976 . . . . . . 7  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  R  e.  Ring )
17 frlmup4.u . . . . . . . 8  |-  U  =  ( R unitVec  I )
1817, 1, 2uvcff 27343 . . . . . . 7  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  U : I --> ( Base `  F ) )
1916, 7, 18syl2anc 642 . . . . . 6  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  U : I --> ( Base `  F ) )
20 ffn 5405 . . . . . 6  |-  ( U : I --> ( Base `  F )  ->  U  Fn  I )
2119, 20syl 15 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  U  Fn  I )
22 frn 5411 . . . . . 6  |-  ( U : I --> ( Base `  F )  ->  ran  U 
C_  ( Base `  F
) )
2319, 22syl 15 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  ran  U  C_  ( Base `  F ) )
24 fnco 5368 . . . . 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 1182 . . . 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 5405 . . . . 5  |-  ( A : I --> C  ->  A  Fn  I )
27263ad2ant3 978 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  A  Fn  I )
2819adantr 451 . . . . . . 7  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  U :
I --> ( Base `  F
) )
2928, 20syl 15 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  U  Fn  I )
30 simpr 447 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  y  e.  I )
31 fvco2 5610 . . . . . 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 642 . . . . 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 958 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  T  e.  LMod )
34 simpl2 959 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  I  e.  X )
358a1i 10 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C )  /\  y  e.  I
)  ->  R  =  (Scalar `  T ) )
36 simpl3 960 . . . . . 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 27354 . . . . 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 2328 . . . 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 5641 . . 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 4857 . . . . 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 2304 . . . 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 2897 . . 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 642 . 2  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E. m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
44 ffun 5407 . . . . 5  |-  ( U : I --> ( Base `  F )  ->  Fun  U )
4519, 44syl 15 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  Fun  U )
46 funcoeqres 5520 . . . . . 6  |-  ( ( Fun  U  /\  (
m  o.  U )  =  A )  -> 
( m  |`  ran  U
)  =  ( A  o.  `' U ) )
4746ex 423 . . . . 5  |-  ( Fun 
U  ->  ( (
m  o.  U )  =  A  ->  (
m  |`  ran  U )  =  ( A  o.  `' U ) ) )
4847ralrimivw 2640 . . . 4  |-  ( Fun 
U  ->  A. m  e.  ( F LMHom  T ) ( ( m  o.  U )  =  A  ->  ( m  |`  ran  U )  =  ( A  o.  `' U
) ) )
4945, 48syl 15 . . 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 2296 . . . . . . 7  |-  (LBasis `  F )  =  (LBasis `  F )
511, 17, 50frlmlbs 27352 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  ran  U  e.  (LBasis `  F
) )
5216, 7, 51syl2anc 642 . . . . 5  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  ran  U  e.  (LBasis `  F ) )
53 eqid 2296 . . . . . 6  |-  ( LSpan `  F )  =  (
LSpan `  F )
542, 50, 53lbssp 15848 . . . . 5  |-  ( ran 
U  e.  (LBasis `  F )  ->  (
( LSpan `  F ) `  ran  U )  =  ( Base `  F
) )
5552, 54syl 15 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( ( LSpan `  F
) `  ran  U )  =  ( Base `  F
) )
562, 53lspextmo 15829 . . . 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 642 . . 3  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E* m  e.  ( F LMHom  T ) ( m  |`  ran  U )  =  ( A  o.  `' U ) )
58 rmoim 2977 . . 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 56 . 2  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  E* m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
60 reu5 2766 . 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 645 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   E*wrmo 2559    C_ wss 3165    e. cmpt 4093   `'ccnv 4704   ran crn 4706    |` cres 4707    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    gsumg cgsu 13417   Ringcrg 15353   LModclmod 15643   LSpanclspn 15744   LMHom clmhm 15792  LBasisclbs 15843   freeLMod cfrlm 27315   unitVec cuvc 27316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lmhm 15795  df-lbs 15844  df-sra 15941  df-rgmod 15942  df-nzr 16026  df-dsmm 27301  df-frlm 27317  df-uvc 27318
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