Theorem igsumvalx 13336

Description: Expand out the substitutions in df-igsum 13206. (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 13206	. . 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 4899	. . . . . . 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 2240	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> dom g = A )
8	7	eqeq1d 2216	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = (/) <-> A = (/) ) )
9		simprl 529	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> w = G )
10	9	fveq2d 5603	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = ( 0g ` G ) )
11		gsumval.z	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2258	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = .0. )
13	12	eqeq2d 2219	. . . . 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 2216	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = ( m ... n ) <-> A = ( m ... n ) ) )
16		eqidd 2208	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> m = m )
17	9	fveq2d 5603	. . . . . . . . . . 11 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = ( +g ` G ) )
18		gsumval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
19	17, 18	eqtr4di 2258	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = .+ )
20	16, 19, 3	seqeq123d 10638	. . . . . . . . 9 |- ( ( ph /\ ( w = G /\ g = F ) ) -> seq m ( ( +g ` w ) , g ) = seq m ( .+ , F ) )
21	20	fveq1d 5601	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( seq m ( ( +g ` w ) , g ) ` n ) = ( seq m ( .+ , F ) ` n ) )
22	21	eqeq2d 2219	. . . . . . 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 2509	. . . . 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 1849	. . . 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 795	. . 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 5273	. 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 2790	. 2 |- ( ph -> G e. _V )
30		gsumvalx.f	. . 3 |- ( ph -> F e. X )
31	30	elexd 2790	. 2 |- ( ph -> F e. _V )
32		unab 3448	. . . 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 3649	. . . . . . 7 |- { .0. } = { x | x = .0. }
34		fn0g 13322	. . . . . . . . . 10 |- 0g Fn _V
35		funfvex 5616	. . . . . . . . . . 11 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
36	35	funfni 5395	. . . . . . . . . 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 2294	. . . . . . . 8 |- ( ph -> .0. e. _V )
39		snexg 4244	. . . . . . . 8 |- ( .0. e. _V -> { .0. } e. _V )
40	38, 39	syl 14	. . . . . . 7 |- ( ph -> { .0. } e. _V )
41	33, 40	eqeltrrid 2295	. . . . . 6 |- ( ph -> { x | x = .0. } e. _V )
42		simpr 110	. . . . . . . 8 |- ( ( A = (/) /\ x = .0. ) -> x = .0. )
43	42	ss2abi 3273	. . . . . . 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 4200	. . . . 5 |- ( ph -> { x | ( A = (/) /\ x = .0. ) } e. _V )
46		zex 9416	. . . . . . 7 |- ZZ e. _V
47	46, 46	ab2rexex 6239	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) } e. _V
48		df-rex 2492	. . . . . . . . . . . 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 9688	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
50		eluzelz 9692	. . . . . . . . . . . . . . . 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 1624	. . . . . . . . . . . 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 1931	. . . . . . . . . . 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 2492	. . . . . . . . . . 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 1624	. . . . . . . 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 2492	. . . . . . . 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 3273	. . . . . 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 4198	. . . . 5 |- { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } e. _V
68		unexg 4508	. . . . 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 2295	. . 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 5269	. . 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 6096	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 710   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   u. cun 3172   C_ wss 3174   (/)c0 3468   {csn 3643   dom cdm 4693   iotacio 5249   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   seqcseq 10629   Basecbs 12947   +g cplusg 13024   0gc0g 13203   gsum_g cgsu 13204

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-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

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-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-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-id 4358  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-recs 6414  df-frec 6500  df-neg 8281  df-inn 9072  df-z 9408  df-uz 9684  df-seqfrec 10630  df-ndx 12950  df-slot 12951  df-base 12953  df-0g 13205  df-igsum 13206

This theorem is referenced by:  igsumval  13337  gsumpropd  13339  gsumpropd2  13340