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Theorem frlmsplit2 18977
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 994 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  R  e.  Ring )
2 simp2 995 . . . . . 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 2454 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  U ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) )
63, 4, 5frlmlss 18958 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
71, 2, 6syl2anc 659 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
8 eqid 2454 . . . . . 6  |-  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  =  ( Base `  (
(ringLMod `  R )  ^s  U
) )
98, 5lssss 17781 . . . . 5  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  U
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  U
) ) )
10 resmpt 5311 . . . . 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 2513 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  F )
14 rlmlmod 18049 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
15 eqid 2454 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  U )  =  ( (ringLMod `  R
)  ^s  U )
16 eqid 2454 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  V )  =  ( (ringLMod `  R
)  ^s  V )
17 eqid 2454 . . . . . . 7  |-  ( Base `  ( (ringLMod `  R
)  ^s  V ) )  =  ( Base `  (
(ringLMod `  R )  ^s  V
) )
18 eqid 2454 . . . . . . 7  |-  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  =  ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )
1915, 16, 8, 17, 18pwssplit3 17905 . . . . . 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 1259 . . . . 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 2454 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  U )s  B )  =  ( ( (ringLMod `  R
)  ^s  U )s  B )
225, 21reslmhm 17896 . . . . 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 659 . . . 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 1015 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (ringLMod `  R )  e.  LMod )
25 simp3 996 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
262, 25ssexd 4584 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
2716pwslmod 17814 . . . . . 6  |-  ( ( (ringLMod `  R )  e.  LMod  /\  V  e.  _V )  ->  ( (ringLMod `  R )  ^s  V )  e.  LMod )
2824, 26, 27syl2anc 659 . . . . 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 2454 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) )
3229, 30, 31frlmlss 18958 . . . . . 6  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
331, 26, 32syl2anc 659 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
3411rneqd 5219 . . . . . 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 2454 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
363, 35, 4frlmbasf 18968 . . . . . . . . . . . 12  |-  ( ( U  e.  X  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
372, 36sylan 469 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
38 simpl3 999 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  V  C_  U )
3937, 38fssresd 5734 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V ) : V --> ( Base `  R ) )
40 fvex 5858 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
41 elmapg 7425 . . . . . . . . . . . 12  |-  ( ( ( Base `  R
)  e.  _V  /\  V  e.  _V )  ->  ( ( x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  <->  ( x  |`  V ) : V --> ( Base `  R
) ) )
4240, 26, 41sylancr 661 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  |`  V )  e.  ( ( Base `  R )  ^m  V
)  <->  ( x  |`  V ) : V --> ( Base `  R )
) )
4342adantr 463 . . . . . . . . . 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
) ) )
4439, 43mpbird 232 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  ( ( Base `  R )  ^m  V
) )
45 eqid 2454 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
463, 45, 4frlmbasfsupp 18966 . . . . . . . . . . 11  |-  ( ( U  e.  X  /\  x  e.  B )  ->  x finSupp  ( 0g `  R ) )
472, 46sylan 469 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  x finSupp  ( 0g `  R ) )
48 fvex 5858 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
4948a1i 11 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( 0g `  R
)  e.  _V )
5047, 49fsuppres 7846 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V ) finSupp 
( 0g `  R
) )
5129, 35, 45, 30frlmelbas 18964 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  (
( x  |`  V )  e.  C  <->  ( (
x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  /\  ( x  |`  V ) finSupp 
( 0g `  R
) ) ) )
521, 26, 51syl2anc 659 . . . . . . . . . 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
) ) ) )
5352adantr 463 . . . . . . . . 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 ) ) ) )
5444, 50, 53mpbir2and 920 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  C )
55 eqid 2454 . . . . . . . 8  |-  ( x  e.  B  |->  ( x  |`  V ) )  =  ( x  e.  B  |->  ( x  |`  V ) )
5654, 55fmptd 6031 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  B  |->  ( x  |`  V )
) : B --> C )
57 frn 5719 . . . . . . 7  |-  ( ( x  e.  B  |->  ( x  |`  V )
) : B --> C  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
5856, 57syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
5934, 58eqsstrd 3523 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  C_  C
)
60 eqid 2454 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  V )s  C )  =  ( ( (ringLMod `  R
)  ^s  V )s  C )
6160, 31reslmhm2b 17898 . . . . 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 ) ) ) )
6228, 33, 59, 61syl3anc 1226 . . . 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 ) ) ) )
6323, 62mpbid 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 ) ) )
6413, 63eqeltrrd 2543 . 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 ) ) )
653, 4frlmpws 18957 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
661, 2, 65syl2anc 659 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
6729, 30frlmpws 18957 . . . 4  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
681, 26, 67syl2anc 659 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
6966, 68oveq12d 6288 . 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 ) ) )
7064, 69eleqtrrd 2545 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497   ran crn 4989    |` cres 4990   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   finSupp cfsupp 7821   Basecbs 14719   ↾s cress 14720   0gc0g 14932    ^s cpws 14939   Ringcrg 17396   LModclmod 17710   LSubSpclss 17776   LMHom clmhm 17863  ringLModcrglmod 18013   freeLMod cfrlm 18953
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-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  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-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-hom 14811  df-cco 14812  df-0g 14934  df-prds 14940  df-pws 14942  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-subg 16400  df-ghm 16467  df-mgp 17340  df-ur 17352  df-ring 17398  df-subrg 17625  df-lmod 17712  df-lss 17777  df-lmhm 17866  df-sra 18016  df-rgmod 18017  df-dsmm 18939  df-frlm 18954
This theorem is referenced by:  frlmsslss  18978
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