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Theorem gsumpropd2 16103
Description: A stronger version of gsumpropd 16101, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 16104. (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:    t, s, F    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
StepHypRef Expression
1 gsumpropd2.f . 2  |-  ( ph  ->  F  e.  V )
2 gsumpropd2.g . 2  |-  ( ph  ->  G  e.  W )
3 gsumpropd2.h . 2  |-  ( ph  ->  H  e.  X )
4 gsumpropd2.b . 2  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
5 gsumpropd2.c . 2  |-  ( (
ph  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  e.  ( Base `  G
) )
6 gsumpropd2.e . 2  |-  ( (
ph  /\  ( s  e.  ( Base `  G
)  /\  t  e.  ( Base `  G )
) )  ->  (
s ( +g  `  G
) t )  =  ( s ( +g  `  H ) t ) )
7 gsumpropd2.n . 2  |-  ( ph  ->  Fun  F )
8 gsumpropd2.r . 2  |-  ( ph  ->  ran  F  C_  ( Base `  G ) )
9 eqid 2454 . 2  |-  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  =  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )
10 eqid 2454 . 2  |-  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  =  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10gsumpropd2lem 16102 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   `'ccnv 4987   ran crn 4989   "cima 4991   Fun wfun 5564   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787    gsumg cgsu 14933
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-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-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-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-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  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-seq 12093  df-0g 14934  df-gsum 14935
This theorem is referenced by:  gsummgmpropd  16104  esumpfinvallem  28306
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