Theorem gsum2dlem1 18369

Description: Lemma 1 for gsum2d 18371. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)

Hypotheses

Ref	Expression
gsum2d.b	|- B = ( Base ` G )
gsum2d.z	|- .0. = ( 0g ` G )
gsum2d.g	|- ( ph -> G e. CMnd )
gsum2d.a	|- ( ph -> A e. V )
gsum2d.r	|- ( ph -> Rel A )
gsum2d.d	|- ( ph -> D e. W )
gsum2d.s	|- ( ph -> dom A C_ D )
gsum2d.f	|- ( ph -> F : A --> B )
gsum2d.w	|- ( ph -> F finSupp .0. )

Assertion

Ref	Expression
gsum2dlem1	|- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )

Distinct variable groups:   j, k, A   j, F, k   j, G, k   ph, j, k   B, j, k   D, j, k   .0. , j, k

Allowed substitution hints:   V( j, k)   W( j, k)

Proof of Theorem gsum2dlem1

Step	Hyp	Ref	Expression
1		gsum2d.b	. 2 |- B = ( Base ` G )
2		gsum2d.z	. 2 |- .0. = ( 0g ` G )
3		gsum2d.g	. 2 |- ( ph -> G e. CMnd )
4		gsum2d.a	. . 3 |- ( ph -> A e. V )
5		imaexg 7103	. . 3 |- ( A e. V -> ( A " { j } ) e. _V )
6	4, 5	syl 17	. 2 |- ( ph -> ( A " { j } ) e. _V )
7		vex 3203	. . . . 5 |- j e. _V
8		vex 3203	. . . . 5 |- k e. _V
9	7, 8	elimasn 5490	. . . 4 |- ( k e. ( A " { j } ) <-> <. j , k >. e. A )
10		df-ov 6653	. . . . 5 |- ( j F k ) = ( F ` <. j , k >. )
11		gsum2d.f	. . . . . 6 |- ( ph -> F : A --> B )
12	11	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ <. j , k >. e. A ) -> ( F ` <. j , k >. ) e. B )
13	10, 12	syl5eqel 2705	. . . 4 |- ( ( ph /\ <. j , k >. e. A ) -> ( j F k ) e. B )
14	9, 13	sylan2b 492	. . 3 |- ( ( ph /\ k e. ( A " { j } ) ) -> ( j F k ) e. B )
15		eqid 2622	. . 3 |- ( k e. ( A " { j } ) |-> ( j F k ) ) = ( k e. ( A " { j } ) |-> ( j F k ) )
16	14, 15	fmptd 6385	. 2 |- ( ph -> ( k e. ( A " { j } ) |-> ( j F k ) ) : ( A " { j } ) --> B )
17		gsum2d.w	. . . . 5 |- ( ph -> F finSupp .0. )
18	17	fsuppimpd 8282	. . . 4 |- ( ph -> ( F supp .0. ) e. Fin )
19		rnfi 8249	. . . 4 |- ( ( F supp .0. ) e. Fin -> ran ( F supp .0. ) e. Fin )
20	18, 19	syl 17	. . 3 |- ( ph -> ran ( F supp .0. ) e. Fin )
21	9	biimpi 206	. . . . . . 7 |- ( k e. ( A " { j } ) -> <. j , k >. e. A )
22	7, 8	opelrn 5357	. . . . . . . 8 |- ( <. j , k >. e. ( F supp .0. ) -> k e. ran ( F supp .0. ) )
23	22	con3i 150	. . . . . . 7 |- ( -. k e. ran ( F supp .0. ) -> -. <. j , k >. e. ( F supp .0. ) )
24	21, 23	anim12i 590	. . . . . 6 |- ( ( k e. ( A " { j } ) /\ -. k e. ran ( F supp .0. ) ) -> ( <. j , k >. e. A /\ -. <. j , k >. e. ( F supp .0. ) ) )
25		eldif 3584	. . . . . 6 |- ( k e. ( ( A " { j } ) \ ran ( F supp .0. ) ) <-> ( k e. ( A " { j } ) /\ -. k e. ran ( F supp .0. ) ) )
26		eldif 3584	. . . . . 6 |- ( <. j , k >. e. ( A \ ( F supp .0. ) ) <-> ( <. j , k >. e. A /\ -. <. j , k >. e. ( F supp .0. ) ) )
27	24, 25, 26	3imtr4i 281	. . . . 5 |- ( k e. ( ( A " { j } ) \ ran ( F supp .0. ) ) -> <. j , k >. e. ( A \ ( F supp .0. ) ) )
28		ssid 3624	. . . . . . . 8 |- ( F supp .0. ) C_ ( F supp .0. )
29	28	a1i 11	. . . . . . 7 |- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) )
30		fvex 6201	. . . . . . . . 9 |- ( 0g ` G ) e. _V
31	2, 30	eqeltri 2697	. . . . . . . 8 |- .0. e. _V
32	31	a1i 11	. . . . . . 7 |- ( ph -> .0. e. _V )
33	11, 29, 4, 32	suppssr 7326	. . . . . 6 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( F ` <. j , k >. ) = .0. )
34	10, 33	syl5eq 2668	. . . . 5 |- ( ( ph /\ <. j , k >. e. ( A \ ( F supp .0. ) ) ) -> ( j F k ) = .0. )
35	27, 34	sylan2 491	. . . 4 |- ( ( ph /\ k e. ( ( A " { j } ) \ ran ( F supp .0. ) ) ) -> ( j F k ) = .0. )
36	35, 6	suppss2 7329	. . 3 |- ( ph -> ( ( k e. ( A " { j } ) |-> ( j F k ) ) supp .0. ) C_ ran ( F supp .0. ) )
37		ssfi 8180	. . 3 |- ( ( ran ( F supp .0. ) e. Fin /\ ( ( k e. ( A " { j } ) |-> ( j F k ) ) supp .0. ) C_ ran ( F supp .0. ) ) -> ( ( k e. ( A " { j } ) |-> ( j F k ) ) supp .0. ) e. Fin )
38	20, 36, 37	syl2anc 693	. 2 |- ( ph -> ( ( k e. ( A " { j } ) |-> ( j F k ) ) supp .0. ) e. Fin )
39	1, 2, 3, 6, 16, 38	gsumcl2 18315	1 |- ( ph -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsum2dlem2  18370  gsum2d  18371