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Theorem frlmsplit2 18672
Description: Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
Hypotheses
Ref Expression
frlmsplit2.y  |-  Y  =  ( R freeLMod  U )
frlmsplit2.z  |-  Z  =  ( R freeLMod  V )
frlmsplit2.b  |-  B  =  ( Base `  Y
)
frlmsplit2.c  |-  C  =  ( Base `  Z
)
frlmsplit2.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
frlmsplit2  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
Distinct variable groups:    x, Y    x, R    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem frlmsplit2
StepHypRef Expression
1 simp1 996 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  R  e.  Ring )
2 simp2 997 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
3 frlmsplit2.y . . . . . . 7  |-  Y  =  ( R freeLMod  U )
4 frlmsplit2.b . . . . . . 7  |-  B  =  ( Base `  Y
)
5 eqid 2467 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  U ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) )
63, 4, 5frlmlss 18651 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
71, 2, 6syl2anc 661 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
8 eqid 2467 . . . . . 6  |-  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  =  ( Base `  (
(ringLMod `  R )  ^s  U
) )
98, 5lssss 17454 . . . . 5  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  U
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  U
) ) )
10 resmpt 5329 . . . . 5  |-  ( B 
C_  ( Base `  (
(ringLMod `  R )  ^s  U
) )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  ( x  e.  B  |->  ( x  |`  V )
) )
117, 9, 103syl 20 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  ( x  e.  B  |->  ( x  |`  V )
) )
12 frlmsplit2.f . . . 4  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
1311, 12syl6eqr 2526 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  F )
14 rlmlmod 17722 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
15 eqid 2467 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  U )  =  ( (ringLMod `  R
)  ^s  U )
16 eqid 2467 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  V )  =  ( (ringLMod `  R
)  ^s  V )
17 eqid 2467 . . . . . . 7  |-  ( Base `  ( (ringLMod `  R
)  ^s  V ) )  =  ( Base `  (
(ringLMod `  R )  ^s  V
) )
18 eqid 2467 . . . . . . 7  |-  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  =  ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )
1915, 16, 8, 17, 18pwssplit3 17578 . . . . . 6  |-  ( ( (ringLMod `  R )  e.  LMod  /\  U  e.  X  /\  V  C_  U
)  ->  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  e.  ( ( (ringLMod `  R
)  ^s  U ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
2014, 19syl3an1 1261 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  |->  ( x  |`  V )
)  e.  ( ( (ringLMod `  R )  ^s  U ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
21 eqid 2467 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  U )s  B )  =  ( ( (ringLMod `  R
)  ^s  U )s  B )
225, 21reslmhm 17569 . . . . 5  |-  ( ( ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  e.  ( ( (ringLMod `  R
)  ^s  U ) LMHom  ( (ringLMod `  R )  ^s  V ) )  /\  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
2320, 7, 22syl2anc 661 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
24143ad2ant1 1017 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (ringLMod `  R )  e.  LMod )
25 simp3 998 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
262, 25ssexd 4600 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
2716pwslmod 17487 . . . . . 6  |-  ( ( (ringLMod `  R )  e.  LMod  /\  V  e.  _V )  ->  ( (ringLMod `  R )  ^s  V )  e.  LMod )
2824, 26, 27syl2anc 661 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
(ringLMod `  R )  ^s  V
)  e.  LMod )
29 frlmsplit2.z . . . . . . 7  |-  Z  =  ( R freeLMod  V )
30 frlmsplit2.c . . . . . . 7  |-  C  =  ( Base `  Z
)
31 eqid 2467 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) )
3229, 30, 31frlmlss 18651 . . . . . 6  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
331, 26, 32syl2anc 661 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
3411rneqd 5236 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  =  ran  ( x  e.  B  |->  ( x  |`  V ) ) )
35 eqid 2467 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
363, 35, 4frlmbasf 18663 . . . . . . . . . . . 12  |-  ( ( U  e.  X  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
372, 36sylan 471 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
38 simpl3 1001 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  V  C_  U )
39 fssres 5757 . . . . . . . . . . 11  |-  ( ( x : U --> ( Base `  R )  /\  V  C_  U )  ->  (
x  |`  V ) : V --> ( Base `  R
) )
4037, 38, 39syl2anc 661 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V ) : V --> ( Base `  R ) )
41 fvex 5882 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
42 elmapg 7445 . . . . . . . . . . . 12  |-  ( ( ( Base `  R
)  e.  _V  /\  V  e.  _V )  ->  ( ( x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  <->  ( x  |`  V ) : V --> ( Base `  R
) ) )
4341, 26, 42sylancr 663 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  |`  V )  e.  ( ( Base `  R )  ^m  V
)  <->  ( x  |`  V ) : V --> ( Base `  R )
) )
4443adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( ( x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  <->  ( x  |`  V ) : V --> ( Base `  R
) ) )
4540, 44mpbird 232 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  ( ( Base `  R )  ^m  V
) )
46 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
473, 46, 4frlmbasfsupp 18660 . . . . . . . . . . 11  |-  ( ( U  e.  X  /\  x  e.  B )  ->  x finSupp  ( 0g `  R ) )
482, 47sylan 471 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  x finSupp  ( 0g `  R ) )
49 fvex 5882 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
5049a1i 11 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( 0g `  R
)  e.  _V )
5148, 50fsuppres 7866 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V ) finSupp 
( 0g `  R
) )
5229, 35, 46, 30frlmelbas 18657 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  (
( x  |`  V )  e.  C  <->  ( (
x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  /\  ( x  |`  V ) finSupp 
( 0g `  R
) ) ) )
531, 26, 52syl2anc 661 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  |`  V )  e.  C  <->  ( (
x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  /\  ( x  |`  V ) finSupp 
( 0g `  R
) ) ) )
5453adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( ( x  |`  V )  e.  C  <->  ( ( x  |`  V )  e.  ( ( Base `  R )  ^m  V
)  /\  ( x  |`  V ) finSupp  ( 0g
`  R ) ) ) )
5545, 51, 54mpbir2and 920 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  C )
56 eqid 2467 . . . . . . . 8  |-  ( x  e.  B  |->  ( x  |`  V ) )  =  ( x  e.  B  |->  ( x  |`  V ) )
5755, 56fmptd 6056 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  B  |->  ( x  |`  V )
) : B --> C )
58 frn 5743 . . . . . . 7  |-  ( ( x  e.  B  |->  ( x  |`  V )
) : B --> C  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
5957, 58syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
6034, 59eqsstrd 3543 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  C_  C
)
61 eqid 2467 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  V )s  C )  =  ( ( (ringLMod `  R
)  ^s  V )s  C )
6261, 31reslmhm2b 17571 . . . . 5  |-  ( ( ( (ringLMod `  R
)  ^s  V )  e.  LMod  /\  C  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  /\  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  C_  C
)  ->  ( (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) )  <->  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) ) )
6328, 33, 60, 62syl3anc 1228 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) )  <->  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) ) )
6423, 63mpbid 210 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) )
6513, 64eqeltrrd 2556 . 2  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( ( ( (ringLMod `  R )  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) )
663, 4frlmpws 18650 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
671, 2, 66syl2anc 661 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
6829, 30frlmpws 18650 . . . 4  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
691, 26, 68syl2anc 661 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
7067, 69oveq12d 6313 . 2  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ( Y LMHom  Z )  =  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) )
7165, 70eleqtrrd 2558 1  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ran crn 5006    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   finSupp cfsupp 7841   Basecbs 14507   ↾s cress 14508   0gc0g 14712    ^s cpws 14719   Ringcrg 17070   LModclmod 17383   LSubSpclss 17449   LMHom clmhm 17536  ringLModcrglmod 17686   freeLMod cfrlm 18646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lmhm 17539  df-sra 17689  df-rgmod 17690  df-dsmm 18632  df-frlm 18647
This theorem is referenced by:  frlmsslss  18673
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