Theorem igsumvalx 13619

Description: Expand out the substitutions in df-igsum 13489. (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 13489	. . 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 533	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> g = F )
4	3	dmeqd 4960	. . . . . . 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 2267	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> dom g = A )
8	7	eqeq1d 2243	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = (/) <-> A = (/) ) )
9		simprl 531	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> w = G )
10	9	fveq2d 5676	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = ( 0g ` G ) )
11		gsumval.z	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2285	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = .0. )
13	12	eqeq2d 2246	. . . . 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 2243	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = ( m ... n ) <-> A = ( m ... n ) ) )
16		eqidd 2235	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> m = m )
17	9	fveq2d 5676	. . . . . . . . . . 11 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = ( +g ` G ) )
18		gsumval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
19	17, 18	eqtr4di 2285	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = .+ )
20	16, 19, 3	seqeq123d 10822	. . . . . . . . 9 |- ( ( ph /\ ( w = G /\ g = F ) ) -> seq m ( ( +g ` w ) , g ) = seq m ( .+ , F ) )
21	20	fveq1d 5674	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( seq m ( ( +g ` w ) , g ) ` n ) = ( seq m ( .+ , F ) ` n ) )
22	21	eqeq2d 2246	. . . . . . 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 2545	. . . . 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 1874	. . . 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 801	. . 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 5337	. 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 2829	. 2 |- ( ph -> G e. _V )
30		gsumvalx.f	. . 3 |- ( ph -> F e. X )
31	30	elexd 2829	. 2 |- ( ph -> F e. _V )
32		unab 3490	. . . 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 3697	. . . . . . 7 |- { .0. } = { x | x = .0. }
34		fn0g 13605	. . . . . . . . . 10 |- 0g Fn _V
35		funfvex 5689	. . . . . . . . . . 11 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
36	35	funfni 5460	. . . . . . . . . 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 2321	. . . . . . . 8 |- ( ph -> .0. e. _V )
39		snexg 4299	. . . . . . . 8 |- ( .0. e. _V -> { .0. } e. _V )
40	38, 39	syl 14	. . . . . . 7 |- ( ph -> { .0. } e. _V )
41	33, 40	eqeltrrid 2322	. . . . . 6 |- ( ph -> { x | x = .0. } e. _V )
42		simpr 110	. . . . . . . 8 |- ( ( A = (/) /\ x = .0. ) -> x = .0. )
43	42	ss2abi 3312	. . . . . . 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 4252	. . . . 5 |- ( ph -> { x | ( A = (/) /\ x = .0. ) } e. _V )
46		zex 9588	. . . . . . 7 |- ZZ e. _V
47	46, 46	ab2rexex 6326	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) } e. _V
48		df-rex 2528	. . . . . . . . . . . 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 9861	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
50		eluzelz 9866	. . . . . . . . . . . . . . . 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 1649	. . . . . . . . . . . 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 1958	. . . . . . . . . . 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 2528	. . . . . . . . . . 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 1649	. . . . . . . 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 2528	. . . . . . . 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 3312	. . . . . 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 4250	. . . . 5 |- { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } e. _V
68		unexg 4566	. . . . 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 2322	. . 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 5333	. . 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 6183	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 716   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   E.wrex 2523   _Vcvv 2815   u. cun 3211   C_ wss 3213   (/)c0 3510   {csn 3691   dom cdm 4751   iotacio 5312   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   e. cmpo 6054   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345   seqcseq 10813   Basecbs 13229   +g cplusg 13307   0gc0g 13486   gsum_g cgsu 13487

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-recs 6538  df-frec 6624  df-neg 8449  df-inn 9240  df-z 9580  df-uz 9857  df-seqfrec 10814  df-ndx 13232  df-slot 13233  df-base 13235  df-0g 13488  df-igsum 13489

This theorem is referenced by:  igsumval  13620  gsumpropd  13622  gsumpropd2  13623