Theorem gsummgmpropd 13467

Description: A stronger version of gsumpropd 13465 if at least one of the involved structures is a magma, see gsumpropd2 13466. (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 2229	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
7		eqid 2229	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
8	6, 7	mgmcl 13432	. . . . 5 |- ( ( G e. Mgm /\ s e. ( Base ` G ) /\ t e. ( Base ` G ) ) -> ( s ( +g ` G ) t ) e. ( Base ` G ) )
9	8	3expib 1230	. . . 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 13466	1 |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3198   ran crn 4724   Fun wfun 5318   ` cfv 5324  (class class class)co 6013   Basecbs 13072   +g cplusg 13150   gsum_g cgsu 13330  Mgmcmgm 13427

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-igsum 13332  df-mgm 13429

This theorem is referenced by: (None)