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Theorem gsumzcl2 17114
Description: Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 17115, because it is not required that  F is a function (actually, the hypothesis always holds for any proper class  F). (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
Hypotheses
Ref Expression
gsumzcl.b  |-  B  =  ( Base `  G
)
gsumzcl.0  |-  .0.  =  ( 0g `  G )
gsumzcl.z  |-  Z  =  (Cntz `  G )
gsumzcl.g  |-  ( ph  ->  G  e.  Mnd )
gsumzcl.a  |-  ( ph  ->  A  e.  V )
gsumzcl.f  |-  ( ph  ->  F : A --> B )
gsumzcl.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzcl2.w  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
Assertion
Ref Expression
gsumzcl2  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)

Proof of Theorem gsumzcl2
Dummy variables  f 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.f . . . . . . 7  |-  ( ph  ->  F : A --> B )
2 gsumzcl.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
3 gsumzcl.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
4 fvex 5858 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
53, 4eqeltri 2538 . . . . . . . 8  |-  .0.  e.  _V
65a1i 11 . . . . . . 7  |-  ( ph  ->  .0.  e.  _V )
7 ssid 3508 . . . . . . . 8  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
87a1i 11 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
91, 2, 6, 8gsumcllem 17111 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
109oveq2d 6286 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
11 gsumzcl.g . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
123gsumz 16204 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
1311, 2, 12syl2anc 659 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
1413adantr 463 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
1510, 14eqtrd 2495 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  .0.  )
16 gsumzcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
1716, 3mndidcl 16137 . . . . . 6  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1811, 17syl 16 . . . . 5  |-  ( ph  ->  .0.  e.  B )
1918adantr 463 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  .0.  e.  B )
2015, 19eqeltrd 2542 . . 3  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  e.  B
)
2120ex 432 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  ->  ( G  gsumg  F )  e.  B
) )
22 eqid 2454 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
23 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
2411adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
252adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
261adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  F : A
--> B )
27 gsumzcl.c . . . . . . . 8  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
2827adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  F  C_  ( Z `  ran  F
) )
29 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( # `  ( F supp  .0.  ) )  e.  NN )
30 f1of1 5797 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
3130ad2antll 726 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-> ( F supp  .0.  ) )
32 suppssdm 6904 . . . . . . . . . 10  |-  ( F supp 
.0.  )  C_  dom  F
33 fdm 5717 . . . . . . . . . . 11  |-  ( F : A --> B  ->  dom  F  =  A )
341, 33syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
3532, 34syl5sseq 3537 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
3635adantr 463 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  A )
37 f1ss 5768 . . . . . . . 8  |-  ( ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )  /\  ( F supp  .0.  )  C_  A )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A )
3831, 36, 37syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-> A
)
39 f1ofo 5805 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
40 forn 5780 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp  .0.  ) )
4139, 40syl 16 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp 
.0.  ) )
4241ad2antll 726 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  f  =  ( F supp  .0.  )
)
437, 42syl5sseqr 3538 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  ran  f )
44 eqid 2454 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
4516, 3, 22, 23, 24, 25, 26, 28, 29, 38, 43, 44gsumval3 17110 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  F )  =  (  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( F supp  .0.  )
) ) )
46 nnuz 11117 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
4729, 46syl6eleq 2552 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( # `  ( F supp  .0.  ) )  e.  ( ZZ>= `  1 )
)
48 f1f 5763 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) --> A )
4938, 48syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) --> A )
50 fco 5723 . . . . . . . . 9  |-  ( ( F : A --> B  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) --> A )  ->  ( F  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  ) ) ) --> B )
5126, 49, 50syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  )
) ) --> B )
5251ffvelrnda 6007 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( F supp 
.0.  ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  ) ) )  /\  k  e.  ( 1 ... ( # `  ( F supp  .0.  )
) ) )  -> 
( ( F  o.  f ) `  k
)  e.  B )
5316, 22mndcl 16128 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  k  e.  B  /\  x  e.  B )  ->  ( k ( +g  `  G ) x )  e.  B )
54533expb 1195 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( k  e.  B  /\  x  e.  B
) )  ->  (
k ( +g  `  G
) x )  e.  B )
5524, 54sylan 469 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( F supp 
.0.  ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  ) ) )  /\  ( k  e.  B  /\  x  e.  B ) )  -> 
( k ( +g  `  G ) x )  e.  B )
5647, 52, 55seqcl 12109 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  (  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( F supp  .0.  )
) )  e.  B
)
5745, 56eqeltrd 2542 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  F )  e.  B )
5857expr 613 . . . 4  |-  ( (
ph  /\  ( # `  ( F supp  .0.  ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ( G  gsumg  F )  e.  B
) )
5958exlimdv 1729 . . 3  |-  ( (
ph  /\  ( # `  ( F supp  .0.  ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  ->  ( G  gsumg  F )  e.  B
) )
6059expimpd 601 . 2  |-  ( ph  ->  ( ( ( # `  ( F supp  .0.  )
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )
)  ->  ( G  gsumg  F )  e.  B ) )
61 gsumzcl2.w . . 3  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
62 fz1f1o 13614 . . 3  |-  ( ( F supp  .0.  )  e.  Fin  ->  ( ( F supp 
.0.  )  =  (/)  \/  ( ( # `  ( F supp  .0.  ) )  e.  NN  /\  E. f 
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
) )
6361, 62syl 16 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  \/  (
( # `  ( F supp 
.0.  ) )  e.  NN  /\  E. f 
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
) )
6421, 60, 63mpjaod 379 1  |-  ( ph  ->  ( G  gsumg  F )  e.  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    C_ wss 3461   (/)c0 3783    |-> cmpt 4497   dom cdm 4988   ran crn 4989    o. ccom 4992   -->wf 5566   -1-1->wf1 5567   -onto->wfo 5568   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   1c1 9482   NNcn 10531   ZZ>=cuz 11082   ...cfz 11675    seqcseq 12089   #chash 12387   Basecbs 14716   +g cplusg 14784   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118  Cntzccntz 16552
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-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  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 12090  df-hash 12388  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-cntz 16554
This theorem is referenced by:  gsumzcl  17115  gsumcl2  17121
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