Theorem igsumvalx 12972

Description: Expand out the substitutions in df-igsum 12870. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
gsumval.b	|- B = ( Base ` G )
gsumval.z	|- .0. = ( 0g ` G )
gsumval.p	|- .+ = ( +g ` G )
gsumval.g	|- ( ph -> G e. V )
gsumvalx.f	|- ( ph -> F e. X )
gsumvalx.a	|- ( ph -> dom F = A )

Assertion

Ref	Expression
igsumvalx	|- ( ph -> ( G gsumg F ) = ( iota x ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )

Distinct variable groups:   x, .+   x, .0.   m, F, n, x   m, G, n, x   ph, m, n, x

Allowed substitution hints:   A( x, m, n)   B( x, m, n)   .+ ( m, n)   V( x, m, n)   X( x, m, n)   .0. ( m, n)

Proof of Theorem igsumvalx

Dummy variables g w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-igsum 12870	. . 3 |- gsumg = ( w e. _V , g e. _V |-> ( iota x ( ( dom g = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) ) ) )
2	1	a1i 9	. 2 |- ( ph -> gsumg = ( w e. _V , g e. _V |-> ( iota x ( ( dom g = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) ) ) ) )
3		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> g = F )
4	3	dmeqd 4864	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> dom g = dom F )
5		gsumvalx.a	. . . . . . . 8 |- ( ph -> dom F = A )
6	5	adantr 276	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> dom F = A )
7	4, 6	eqtrd 2226	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> dom g = A )
8	7	eqeq1d 2202	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = (/) <-> A = (/) ) )
9		simprl 529	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> w = G )
10	9	fveq2d 5558	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = ( 0g ` G ) )
11		gsumval.z	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2244	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = .0. )
13	12	eqeq2d 2205	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( x = ( 0g ` w ) <-> x = .0. ) )
14	8, 13	anbi12d 473	. . . 4 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( ( dom g = (/) /\ x = ( 0g ` w ) ) <-> ( A = (/) /\ x = .0. ) ) )
15	7	eqeq1d 2202	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = ( m ... n ) <-> A = ( m ... n ) ) )
16		eqidd 2194	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> m = m )
17	9	fveq2d 5558	. . . . . . . . . . 11 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = ( +g ` G ) )
18		gsumval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
19	17, 18	eqtr4di 2244	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = .+ )
20	16, 19, 3	seqeq123d 10527	. . . . . . . . 9 |- ( ( ph /\ ( w = G /\ g = F ) ) -> seq m ( ( +g ` w ) , g ) = seq m ( .+ , F ) )
21	20	fveq1d 5556	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( seq m ( ( +g ` w ) , g ) ` n ) = ( seq m ( .+ , F ) ` n ) )
22	21	eqeq2d 2205	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( x = ( seq m ( ( +g ` w ) , g ) ` n ) <-> x = ( seq m ( .+ , F ) ` n ) ) )
23	15, 22	anbi12d 473	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) <-> ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
24	23	rexbidv 2495	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( E. n e. ( ZZ>= ` m ) ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) <-> E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
25	24	exbidv 1836	. . . 4 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( E. m E. n e. ( ZZ>= ` m ) ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) <-> E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
26	14, 25	orbi12d 794	. . 3 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( ( ( dom g = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) ) <-> ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
27	26	iotabidv 5237	. 2 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( iota x ( ( dom g = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom g = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , g ) ` n ) ) ) ) = ( iota x ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
28		gsumval.g	. . 3 |- ( ph -> G e. V )
29	28	elexd 2773	. 2 |- ( ph -> G e. _V )
30		gsumvalx.f	. . 3 |- ( ph -> F e. X )
31	30	elexd 2773	. 2 |- ( ph -> F e. _V )
32		unab 3426	. . . 4 |- ( { x | ( A = (/) /\ x = .0. ) } u. { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } ) = { x | ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) }
33		df-sn 3624	. . . . . . 7 |- { .0. } = { x | x = .0. }
34		fn0g 12958	. . . . . . . . . 10 |- 0g Fn _V
35		funfvex 5571	. . . . . . . . . . 11 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
36	35	funfni 5354	. . . . . . . . . 10 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
37	34, 29, 36	sylancr 414	. . . . . . . . 9 |- ( ph -> ( 0g ` G ) e. _V )
38	11, 37	eqeltrid 2280	. . . . . . . 8 |- ( ph -> .0. e. _V )
39		snexg 4213	. . . . . . . 8 |- ( .0. e. _V -> { .0. } e. _V )
40	38, 39	syl 14	. . . . . . 7 |- ( ph -> { .0. } e. _V )
41	33, 40	eqeltrrid 2281	. . . . . 6 |- ( ph -> { x | x = .0. } e. _V )
42		simpr 110	. . . . . . . 8 |- ( ( A = (/) /\ x = .0. ) -> x = .0. )
43	42	ss2abi 3251	. . . . . . 7 |- { x | ( A = (/) /\ x = .0. ) } C_ { x | x = .0. }
44	43	a1i 9	. . . . . 6 |- ( ph -> { x | ( A = (/) /\ x = .0. ) } C_ { x | x = .0. } )
45	41, 44	ssexd 4169	. . . . 5 |- ( ph -> { x | ( A = (/) /\ x = .0. ) } e. _V )
46		zex 9326	. . . . . . 7 |- ZZ e. _V
47	46, 46	ab2rexex 6183	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) } e. _V
48		df-rex 2478	. . . . . . . . . . . 12 |- ( E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> E. n ( n e. ( ZZ>= ` m ) /\ ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
49		eluzel2 9597	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
50		eluzelz 9601	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> n e. ZZ )
51	49, 50	jca 306	. . . . . . . . . . . . . . 15 |- ( n e. ( ZZ>= ` m ) -> ( m e. ZZ /\ n e. ZZ ) )
52		simpr 110	. . . . . . . . . . . . . . 15 |- ( ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> x = ( seq m ( .+ , F ) ` n ) )
53	51, 52	anim12i 338	. . . . . . . . . . . . . 14 |- ( ( n e. ( ZZ>= ` m ) /\ ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( ( m e. ZZ /\ n e. ZZ ) /\ x = ( seq m ( .+ , F ) ` n ) ) )
54		anass 401	. . . . . . . . . . . . . 14 |- ( ( ( m e. ZZ /\ n e. ZZ ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
55	53, 54	sylib 122	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` m ) /\ ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
56	55	eximi 1611	. . . . . . . . . . . 12 |- ( E. n ( n e. ( ZZ>= ` m ) /\ ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> E. n ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
57	48, 56	sylbi 121	. . . . . . . . . . 11 |- ( E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> E. n ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
58		19.42v 1918	. . . . . . . . . . 11 |- ( E. n ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) <-> ( m e. ZZ /\ E. n ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
59	57, 58	sylib 122	. . . . . . . . . 10 |- ( E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> ( m e. ZZ /\ E. n ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
60		df-rex 2478	. . . . . . . . . . 11 |- ( E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) <-> E. n ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) )
61	60	anbi2i 457	. . . . . . . . . 10 |- ( ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) ) <-> ( m e. ZZ /\ E. n ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
62	59, 61	sylibr 134	. . . . . . . . 9 |- ( E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) ) )
63	62	eximi 1611	. . . . . . . 8 |- ( E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> E. m ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) ) )
64		df-rex 2478	. . . . . . . 8 |- ( E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) <-> E. m ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) ) )
65	63, 64	sylibr 134	. . . . . . 7 |- ( E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) )
66	65	ss2abi 3251	. . . . . 6 |- { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } C_ { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) }
67	47, 66	ssexi 4167	. . . . 5 |- { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } e. _V
68		unexg 4474	. . . . 5 |- ( ( { x | ( A = (/) /\ x = .0. ) } e. _V /\ { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } e. _V ) -> ( { x | ( A = (/) /\ x = .0. ) } u. { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } ) e. _V )
69	45, 67, 68	sylancl 413	. . . 4 |- ( ph -> ( { x | ( A = (/) /\ x = .0. ) } u. { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } ) e. _V )
70	32, 69	eqeltrrid 2281	. . 3 |- ( ph -> { x | ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) } e. _V )
71		iotaexab 5233	. . 3 |- ( { x | ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) } e. _V -> ( iota x ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) e. _V )
72	70, 71	syl 14	. 2 |- ( ph -> ( iota x ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) e. _V )
73	2, 27, 29, 31, 72	ovmpod 6046	1 |- ( ph -> ( G gsumg F ) = ( iota x ( ( A = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   E.wrex 2473   _Vcvv 2760   u. cun 3151   C_ wss 3153   (/)c0 3446   {csn 3618   dom cdm 4659   iotacio 5213   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   Basecbs 12618   +g cplusg 12695   0gc0g 12867   gsum_g cgsu 12868

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-recs 6358  df-frec 6444  df-neg 8193  df-inn 8983  df-z 9318  df-uz 9593  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-igsum 12870

This theorem is referenced by:  igsumval  12973  gsumpropd  12975  gsumpropd2  12976