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Theorem gsummhm2OLD 17163
Description: Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) Obsolete version of gsummhm2 17162 as of 6-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsummhm2OLD.b  |-  B  =  ( Base `  G
)
gsummhm2OLD.z  |-  .0.  =  ( 0g `  G )
gsummhm2OLD.g  |-  ( ph  ->  G  e. CMnd )
gsummhm2OLD.h  |-  ( ph  ->  H  e.  Mnd )
gsummhm2OLD.a  |-  ( ph  ->  A  e.  V )
gsummhm2OLD.k  |-  ( ph  ->  ( x  e.  B  |->  C )  e.  ( G MndHom  H ) )
gsummhm2OLD.f  |-  ( (
ph  /\  k  e.  A )  ->  X  e.  B )
gsummhm2OLD.w  |-  ( ph  ->  ( `' ( k  e.  A  |->  X )
" ( _V  \  {  .0.  } ) )  e.  Fin )
gsummhm2OLD.1  |-  ( x  =  X  ->  C  =  D )
gsummhm2OLD.2  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )
Assertion
Ref Expression
gsummhm2OLD  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Distinct variable groups:    x, k, A    B, k, x    C, k    x, D    x, E    ph, k    x, G    x, H    x, X
Allowed substitution hints:    ph( x)    C( x)    D( k)    E( k)    G( k)    H( k)    V( x, k)    X( k)    .0. ( x, k)

Proof of Theorem gsummhm2OLD
StepHypRef Expression
1 gsummhm2OLD.b . . 3  |-  B  =  ( Base `  G
)
2 gsummhm2OLD.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsummhm2OLD.g . . 3  |-  ( ph  ->  G  e. CMnd )
4 gsummhm2OLD.h . . 3  |-  ( ph  ->  H  e.  Mnd )
5 gsummhm2OLD.a . . 3  |-  ( ph  ->  A  e.  V )
6 gsummhm2OLD.k . . 3  |-  ( ph  ->  ( x  e.  B  |->  C )  e.  ( G MndHom  H ) )
7 gsummhm2OLD.f . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  X  e.  B )
8 eqid 2454 . . . 4  |-  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X )
97, 8fmptd 6031 . . 3  |-  ( ph  ->  ( k  e.  A  |->  X ) : A --> B )
10 gsummhm2OLD.w . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  X )
" ( _V  \  {  .0.  } ) )  e.  Fin )
111, 2, 3, 4, 5, 6, 9, 10gsummhmOLD 17161 . 2  |-  ( ph  ->  ( H  gsumg  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) ) )  =  ( ( x  e.  B  |->  C ) `
 ( G  gsumg  ( k  e.  A  |->  X ) ) ) )
12 eqidd 2455 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X ) )
13 eqidd 2455 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C ) )
14 gsummhm2OLD.1 . . . 4  |-  ( x  =  X  ->  C  =  D )
157, 12, 13, 14fmptco 6040 . . 3  |-  ( ph  ->  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) )  =  ( k  e.  A  |->  D ) )
1615oveq2d 6286 . 2  |-  ( ph  ->  ( H  gsumg  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) ) )  =  ( H  gsumg  ( k  e.  A  |->  D ) ) )
171, 2, 3, 5, 9, 10gsumclOLD 17128 . . 3  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B )
18 eqid 2454 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
191, 18mhmf 16173 . . . . . 6  |-  ( ( x  e.  B  |->  C )  e.  ( G MndHom  H )  ->  (
x  e.  B  |->  C ) : B --> ( Base `  H ) )
206, 19syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  B  |->  C ) : B --> ( Base `  H )
)
21 eqid 2454 . . . . . 6  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
2221fmpt 6028 . . . . 5  |-  ( A. x  e.  B  C  e.  ( Base `  H
)  <->  ( x  e.  B  |->  C ) : B --> ( Base `  H
) )
2320, 22sylibr 212 . . . 4  |-  ( ph  ->  A. x  e.  B  C  e.  ( Base `  H ) )
24 gsummhm2OLD.2 . . . . . 6  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )
2524eleq1d 2523 . . . . 5  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  ( C  e.  ( Base `  H
)  <->  E  e.  ( Base `  H ) ) )
2625rspcv 3203 . . . 4  |-  ( ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B  ->  ( A. x  e.  B  C  e.  ( Base `  H
)  ->  E  e.  ( Base `  H )
) )
2717, 23, 26sylc 60 . . 3  |-  ( ph  ->  E  e.  ( Base `  H ) )
2824, 21fvmptg 5929 . . 3  |-  ( ( ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B  /\  E  e.  ( Base `  H
) )  ->  (
( x  e.  B  |->  C ) `  ( G  gsumg  ( k  e.  A  |->  X ) ) )  =  E )
2917, 27, 28syl2anc 659 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C ) `  ( G  gsumg  ( k  e.  A  |->  X ) ) )  =  E )
3011, 16, 293eqtr3d 2503 1  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    \ cdif 3458   {csn 4016    |-> cmpt 4497   `'ccnv 4987   "cima 4991    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   Basecbs 14719   0gc0g 14932    gsumg cgsu 14933   Mndcmnd 16121   MndHom cmhm 16166  CMndccmn 17000
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-se 4828  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-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311  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-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-cntz 16557  df-cmn 17002
This theorem is referenced by:  prdsgsumOLD  17206  gsummulc1OLD  17452  gsummulc2OLD  17453  gsumvsmulOLD  17773
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