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Theorem gsumpropd2 12976
Description: A stronger version of gsumpropd 12975, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 12977. (Contributed by Thierry Arnoux, 28-Jun-2017.)
Hypotheses
Ref Expression
gsumpropd2.f  |-  ( ph  ->  F  e.  V )
gsumpropd2.g  |-  ( ph  ->  G  e.  W )
gsumpropd2.h  |-  ( ph  ->  H  e.  X )
gsumpropd2.b  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
gsumpropd2.c  |-  ( (
ph  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  e.  ( Base `  G
) )
gsumpropd2.e  |-  ( (
ph  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  =  ( s ( +g  `  H ) t ) )
gsumpropd2.n  |-  ( ph  ->  Fun  F )
gsumpropd2.r  |-  ( ph  ->  ran  F  C_  ( Base `  G ) )
Assertion
Ref Expression
gsumpropd2  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Distinct variable groups:    F, s, t    G, s, t    H, s, t    ph, s, t
Allowed substitution hints:    V( t, s)    W( t, s)    X( t, s)

Proof of Theorem gsumpropd2
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2194 . . . . . . 7  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
2 gsumpropd2.b . . . . . . 7  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
3 gsumpropd2.g . . . . . . 7  |-  ( ph  ->  G  e.  W )
4 gsumpropd2.h . . . . . . 7  |-  ( ph  ->  H  e.  X )
5 gsumpropd2.e . . . . . . 7  |-  ( (
ph  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  =  ( s ( +g  `  H ) t ) )
61, 2, 3, 4, 5grpidpropdg 12957 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
76eqeq2d 2205 . . . . 5  |-  ( ph  ->  ( x  =  ( 0g `  G )  <-> 
x  =  ( 0g
`  H ) ) )
87anbi2d 464 . . . 4  |-  ( ph  ->  ( ( dom  F  =  (/)  /\  x  =  ( 0g `  G
) )  <->  ( dom  F  =  (/)  /\  x  =  ( 0g `  H ) ) ) )
9 simprl 529 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  ->  n  e.  ( ZZ>= `  m )
)
10 gsumpropd2.r . . . . . . . . . . . 12  |-  ( ph  ->  ran  F  C_  ( Base `  G ) )
1110ad2antrr 488 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  ran  F  C_  ( Base `  G )
)
12 gsumpropd2.n . . . . . . . . . . . . 13  |-  ( ph  ->  Fun  F )
1312ad2antrr 488 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  Fun  F )
14 simpr 110 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  s  e.  ( m ... n
) )
15 simplrr 536 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  dom  F  =  ( m ... n
) )
1614, 15eleqtrrd 2273 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  s  e.  dom  F )
17 fvelrn 5689 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  s  e.  dom  F )  -> 
( F `  s
)  e.  ran  F
)
1813, 16, 17syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  ( F `  s )  e.  ran  F )
1911, 18sseldd 3180 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  s  e.  ( m ... n ) )  ->  ( F `  s )  e.  (
Base `  G )
)
20 gsumpropd2.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  V )
2120adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  ->  F  e.  V )
22 plusgslid 12730 . . . . . . . . . . . . 13  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
2322slotex 12645 . . . . . . . . . . . 12  |-  ( G  e.  W  ->  ( +g  `  G )  e. 
_V )
243, 23syl 14 . . . . . . . . . . 11  |-  ( ph  ->  ( +g  `  G
)  e.  _V )
2524adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  ->  ( +g  `  G )  e. 
_V )
2622slotex 12645 . . . . . . . . . . . 12  |-  ( H  e.  X  ->  ( +g  `  H )  e. 
_V )
274, 26syl 14 . . . . . . . . . . 11  |-  ( ph  ->  ( +g  `  H
)  e.  _V )
2827adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  ->  ( +g  `  H )  e. 
_V )
29 gsumpropd2.c . . . . . . . . . . 11  |-  ( (
ph  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  e.  ( Base `  G
) )
3029adantlr 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  e.  ( Base `  G
) )
315adantlr 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  =  ( s ( +g  `  H ) t ) )
329, 19, 21, 25, 28, 30, 31seqfeq4g 10602 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  ->  (  seq m ( ( +g  `  G ) ,  F
) `  n )  =  (  seq m
( ( +g  `  H
) ,  F ) `
 n ) )
3332eqeq2d 2205 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  m )  /\  dom  F  =  ( m ... n ) ) )  ->  (
x  =  (  seq m ( ( +g  `  G ) ,  F
) `  n )  <->  x  =  (  seq m
( ( +g  `  H
) ,  F ) `
 n ) ) )
3433anassrs 400 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  dom  F  =  ( m ... n
) )  ->  (
x  =  (  seq m ( ( +g  `  G ) ,  F
) `  n )  <->  x  =  (  seq m
( ( +g  `  H
) ,  F ) `
 n ) ) )
3534pm5.32da 452 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  m )
)  ->  ( ( dom  F  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  G ) ,  F
) `  n )
)  <->  ( dom  F  =  ( m ... n )  /\  x  =  (  seq m
( ( +g  `  H
) ,  F ) `
 n ) ) ) )
3635rexbidva 2491 . . . . 5  |-  ( ph  ->  ( E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  G ) ,  F ) `  n ) )  <->  E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
3736exbidv 1836 . . . 4  |-  ( ph  ->  ( E. m E. n  e.  ( ZZ>= `  m ) ( dom 
F  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  G ) ,  F
) `  n )
)  <->  E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
388, 37orbi12d 794 . . 3  |-  ( ph  ->  ( ( ( dom 
F  =  (/)  /\  x  =  ( 0g `  G ) )  \/ 
E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  G ) ,  F ) `  n ) ) )  <-> 
( ( dom  F  =  (/)  /\  x  =  ( 0g `  H
) )  \/  E. m E. n  e.  (
ZZ>= `  m ) ( dom  F  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) ) ) )
3938iotabidv 5237 . 2  |-  ( ph  ->  ( iota x ( ( dom  F  =  (/)  /\  x  =  ( 0g `  G ) )  \/  E. m E. n  e.  ( ZZ>=
`  m ) ( dom  F  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  G ) ,  F ) `  n ) ) ) )  =  ( iota
x ( ( dom 
F  =  (/)  /\  x  =  ( 0g `  H ) )  \/ 
E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) ) ) )
40 eqid 2193 . . 3  |-  ( Base `  G )  =  (
Base `  G )
41 eqid 2193 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
42 eqid 2193 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
43 eqidd 2194 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
4440, 41, 42, 3, 20, 43igsumvalx 12972 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( iota x ( ( dom  F  =  (/)  /\  x  =  ( 0g
`  G ) )  \/  E. m E. n  e.  ( ZZ>= `  m ) ( dom 
F  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  G ) ,  F
) `  n )
) ) ) )
45 eqid 2193 . . 3  |-  ( Base `  H )  =  (
Base `  H )
46 eqid 2193 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
47 eqid 2193 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
4845, 46, 47, 4, 20, 43igsumvalx 12972 . 2  |-  ( ph  ->  ( H  gsumg  F )  =  ( iota x ( ( dom  F  =  (/)  /\  x  =  ( 0g
`  H ) )  \/  E. m E. n  e.  ( ZZ>= `  m ) ( dom 
F  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  H ) ,  F
) `  n )
) ) ) )
4939, 44, 483eqtr4d 2236 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364   E.wex 1503    e. wcel 2164   E.wrex 2473   _Vcvv 2760    C_ wss 3153   (/)c0 3446   dom cdm 4659   ran crn 4660   iotacio 5213   Fun wfun 5248   ` cfv 5254  (class class class)co 5918   ZZ>=cuz 9592   ...cfz 10074    seqcseq 10518   Basecbs 12618   +g cplusg 12695   0gc0g 12867    gsumg cgsu 12868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870
This theorem is referenced by:  gsummgmpropd  12977
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