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Theorem frlmup4 27221
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 2435 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
3 frlmup4.c . . . 4  |-  C  =  ( Base `  T
)
4 eqid 2435 . . . 4  |-  ( .s
`  T )  =  ( .s `  T
)
5 eqid 2435 . . . 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 27218 . . 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 6098 . . . . . . 7  |-  ( T 
gsumg  ( x  o F
( .s `  T
) A ) )  e.  _V
1312, 5fnmpti 5565 . . . . . 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 15950 . . . . . . . 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 27208 . . . . . . 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 5583 . . . . . 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 5589 . . . . . 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 5545 . . . . 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 5583 . . . . 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 5790 . . . . . 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 27219 . . . . 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 2467 . . . 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 5822 . . 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 5022 . . . . 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 2443 . . . 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 3044 . . 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 5585 . . . . 5  |-  ( U : I --> ( Base `  F )  ->  Fun  U )
4519, 44syl 16 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  ->  Fun  U )
46 funcoeqres 5698 . . . . . 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 2782 . . . 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 2435 . . . . . . 7  |-  (LBasis `  F )  =  (LBasis `  F )
511, 17, 50frlmlbs 27217 . . . . . 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 2435 . . . . . 6  |-  ( LSpan `  F )  =  (
LSpan `  F )
542, 50, 53lbssp 16143 . . . . 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 16124 . . . 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 3125 . . 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 2913 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699   E*wrmo 2700    C_ wss 3312    e. cmpt 4258   `'ccnv 4869   ran crn 4871    |` cres 4872    o. ccom 4874   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461  Scalarcsca 13524   .scvsca 13525    gsumg cgsu 13716   Ringcrg 15652   LModclmod 15942   LSpanclspn 16039   LMHom clmhm 16087  LBasisclbs 16138   freeLMod cfrlm 27180   unitVec cuvc 27181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-prds 13663  df-pws 13665  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lmhm 16090  df-lbs 16139  df-sra 16236  df-rgmod 16237  df-nzr 16321  df-dsmm 27166  df-frlm 27182  df-uvc 27183
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