Theorem gsummgmpropd 13341

Description: A stronger version of gsumpropd 13339 if at least one of the involved structures is a magma, see gsumpropd2 13340. (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 2207	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
7		eqid 2207	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
8	6, 7	mgmcl 13306	. . . . 5 |- ( ( G e. Mgm /\ s e. ( Base ` G ) /\ t e. ( Base ` G ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
9	8	3expib 1209	. . . 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 13340	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   C_ wss 3174   ran crn 4694   Fun wfun 5284   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   gsum_g cgsu 13204  Mgmcmgm 13301

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-igsum 13206  df-mgm 13303

This theorem is referenced by: (None)