Theorem gsum2dfi 14273

Description: Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 6-Oct-2026.)

Hypotheses

Ref	Expression
gsum2d.b	|- B = ( Base ` G )
gsum2d.z	|- .0. = ( 0g ` G )
gsum2d.g	|- ( ph -> G e. CMnd )
gsum2dfi.a	|- ( ph -> A e. Fin )
gsum2d.r	|- ( ph -> Rel A )
gsum2dfi.d	|- ( ph -> D e. Fin )
gsum2d.s	|- ( ph -> dom A C_ D )
gsum2dfi.dm	|- ( ph -> dom A e. Fin )
gsum2d.f	|- ( ph -> F : A --> B )
gsum2dfi.fi	|- ( ( ph /\ j e. D ) -> ( A " { j } ) e. Fin )

Assertion

Ref	Expression
gsum2dfi	|- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )

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

Proof of Theorem gsum2dfi

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsum2d.b	. . 3 |- B = ( Base ` G )
2		gsum2d.z	. . 3 |- .0. = ( 0g ` G )
3		gsum2d.g	. . 3 |- ( ph -> G e. CMnd )
4		gsum2dfi.a	. . 3 |- ( ph -> A e. Fin )
5		gsum2d.r	. . 3 |- ( ph -> Rel A )
6		gsum2dfi.d	. . 3 |- ( ph -> D e. Fin )
7		gsum2d.s	. . 3 |- ( ph -> dom A C_ D )
8		gsum2dfi.dm	. . 3 |- ( ph -> dom A e. Fin )
9		gsum2d.f	. . 3 |- ( ph -> F : A --> B )
10		gsum2dfi.fi	. . 3 |- ( ( ph /\ j e. D ) -> ( A " { j } ) e. Fin )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	gsum2dfilem2 14272	. 2 |- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. dom A |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
12	7	resmptd 5117	. . 3 |- ( ph -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) |` dom A ) = ( j e. dom A |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) )
13	12	oveq2d 6104	. 2 |- ( ph -> ( G gsumg ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) |` dom A ) ) = ( G gsumg ( j e. dom A |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
14	1, 2, 3, 9, 10	gsum2dfilem1 14271	. . . 4 |- ( ( ph /\ j e. D ) -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) e. B )
15	14	fmpttd 5866	. . 3 |- ( ph -> ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) : D --> B )
16		eqid 2238	. . . . . 6 |- ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) = ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) )
17		sneq 3720	. . . . . . . . 9 |- ( j = q -> { j } = { q } )
18	17	imaeq2d 5129	. . . . . . . 8 |- ( j = q -> ( A " { j } ) = ( A " { q } ) )
19		oveq1 6095	. . . . . . . 8 |- ( j = q -> ( j F k ) = ( q F k ) )
20	18, 19	mpteq12dv 4214	. . . . . . 7 |- ( j = q -> ( k e. ( A " { j } ) |-> ( j F k ) ) = ( k e. ( A " { q } ) |-> ( q F k ) ) )
21	20	oveq2d 6104	. . . . . 6 |- ( j = q -> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) = ( G gsumg ( k e. ( A " { q } ) |-> ( q F k ) ) ) )
22		eldifi 3351	. . . . . . 7 |- ( q e. ( D \ dom A ) -> q e. D )
23	22	adantl 277	. . . . . 6 |- ( ( ph /\ q e. ( D \ dom A ) ) -> q e. D )
24		eldifn 3352	. . . . . . . . . . 11 |- ( q e. ( D \ dom A ) -> -. q e. dom A )
25	24	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ q e. ( D \ dom A ) ) -> -. q e. dom A )
26		ndmima 5167	. . . . . . . . . . . 12 |- ( -. q e. dom A -> ( A " { q } ) = (/) )
27	26	mpteq1d 4217	. . . . . . . . . . 11 |- ( -. q e. dom A -> ( k e. ( A " { q } ) |-> ( q F k ) ) = ( k e. (/) |-> ( q F k ) ) )
28		mpt0 5514	. . . . . . . . . . 11 |- ( k e. (/) |-> ( q F k ) ) = (/)
29	27, 28	eqtrdi 2287	. . . . . . . . . 10 |- ( -. q e. dom A -> ( k e. ( A " { q } ) |-> ( q F k ) ) = (/) )
30	25, 29	syl 14	. . . . . . . . 9 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( k e. ( A " { q } ) |-> ( q F k ) ) = (/) )
31	30	oveq2d 6104	. . . . . . . 8 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( G gsumg ( k e. ( A " { q } ) |-> ( q F k ) ) ) = ( G gsumg (/) ) )
32		gsum0cmn 14244	. . . . . . . . . . 11 |- ( G e. CMnd -> ( G gsumg (/) ) = ( 0g ` G ) )
33	3, 32	syl 14	. . . . . . . . . 10 |- ( ph -> ( G gsumg (/) ) = ( 0g ` G ) )
34	33, 2	eqtr4di 2289	. . . . . . . . 9 |- ( ph -> ( G gsumg (/) ) = .0. )
35	34	adantr 276	. . . . . . . 8 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( G gsumg (/) ) = .0. )
36	31, 35	eqtrd 2271	. . . . . . 7 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( G gsumg ( k e. ( A " { q } ) |-> ( q F k ) ) ) = .0. )
37	3	cmnmndd 14201	. . . . . . . . 9 |- ( ph -> G e. Mnd )
38	1, 2	mndidcl 13802	. . . . . . . . 9 |- ( G e. Mnd -> .0. e. B )
39	37, 38	syl 14	. . . . . . . 8 |- ( ph -> .0. e. B )
40	39	adantr 276	. . . . . . 7 |- ( ( ph /\ q e. ( D \ dom A ) ) -> .0. e. B )
41	36, 40	eqeltrd 2315	. . . . . 6 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( G gsumg ( k e. ( A " { q } ) |-> ( q F k ) ) ) e. B )
42	16, 21, 23, 41	fvmptd3 5802	. . . . 5 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ` q ) = ( G gsumg ( k e. ( A " { q } ) |-> ( q F k ) ) ) )
43	42, 36	eqtrd 2271	. . . 4 |- ( ( ph /\ q e. ( D \ dom A ) ) -> ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ` q ) = .0. )
44	43	ralrimiva 2623	. . 3 |- ( ph -> A. q e. ( D \ dom A ) ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ` q ) = .0. )
45	1, 2, 3, 6, 15, 44, 8, 7	gsumresfi 14263	. 2 |- ( ph -> ( G gsumg ( ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) |` dom A ) ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )
46	11, 13, 45	3eqtr2d 2277	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( j e. D |-> ( G gsumg ( k e. ( A " { j } ) |-> ( j F k ) ) ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   \ cdif 3217   C_ wss 3220   (/)c0 3520   {csn 3709   |-> cmpt 4193   dom cdm 4776   |` cres 4778   "cima 4779   Rel wrel 4781   -->wf 5376   ` cfv 5380  (class class class)co 6088   Fincfn 7026   Basecbs 13410   0gc0g 13669   Mndcmnd 13788  CMndccmn 14177   gsum_g cgsu 14240

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-minusg 13868  df-mulg 13982  df-cmn 14179  df-gsumfi 14241

This theorem is used by:  gsum2d2fi  14274