Theorem gsummgmpropd 13657

Description: A stronger version of gsumpropd 13655 if at least one of the involved structures is a magma, see gsumpropd2 13656. (Contributed by AV, 31-Jan-2020.)

Hypotheses

Ref	Expression
gsummgmpropd.f	|- ( ph -> F e. V )
gsummgmpropd.g	|- ( ph -> G e. W )
gsummgmpropd.h	|- ( ph -> H e. X )
gsummgmpropd.b	|- ( ph -> ( Base ` G ) = ( Base ` H ) )
gsummgmpropd.m	|- ( ph -> G e. Mgm )
gsummgmpropd.e	|- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
gsummgmpropd.n	|- ( ph -> Fun F )
gsummgmpropd.r	|- ( ph -> ran F C_ ( Base ` G ) )

Assertion

Ref	Expression
gsummgmpropd	|- ( 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 gsummgmpropd

Step	Hyp	Ref	Expression
1		gsummgmpropd.f	. 2 |- ( ph -> F e. V )
2		gsummgmpropd.g	. 2 |- ( ph -> G e. W )
3		gsummgmpropd.h	. 2 |- ( ph -> H e. X )
4		gsummgmpropd.b	. 2 |- ( ph -> ( Base ` G ) = ( Base ` H ) )
5		gsummgmpropd.m	. . . 4 |- ( ph -> G e. Mgm )
6		eqid 2234	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
7		eqid 2234	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
8	6, 7	mgmcl 13622	. . . . 5 |- ( ( G e. Mgm /\ s e. ( Base ` G ) /\ t e. ( Base ` G ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
9	8	3expib 1233	. . . 4 |- ( G e. Mgm -> ( ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) ) )
10	5, 9	syl 14	. . 3 |- ( ph -> ( ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) ) )
11	10	imp 124	. 2 |- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
12		gsummgmpropd.e	. 2 |- ( ( ph /\ ( s e. ( Base ` G ) /\ t e. ( Base ` G ) ) ) -> ( s ( +g ` G ) t ) = ( s ( +g ` H ) t ) )
13		gsummgmpropd.n	. 2 |- ( ph -> Fun F )
14		gsummgmpropd.r	. 2 |- ( ph -> ran F C_ ( Base ` G ) )
15	1, 2, 3, 4, 11, 12, 13, 14	gsumpropd2 13656	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214   ran crn 4755   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   gsum_g cgsu 13554  Mgmcmgm 13617

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556  df-mgm 13619

This theorem is referenced by: (None)