Theorem gzsumvalx 13686

Description: Expand out the substitutions in df-gzsum 13590. (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
gzsumvalx	|- ( ph -> ( G gzsumgz 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 gzsumvalx

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

Step	Hyp	Ref	Expression
1		df-gzsum 13590	. . 3 |- gzsumgz = ( 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 -> gzsumgz = ( 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 537	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> g = F )
4	3	dmeqd 4978	. . . . . . 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 2271	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> dom g = A )
8	7	eqeq1d 2247	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = (/) <-> A = (/) ) )
9		simprl 535	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> w = G )
10	9	fveq2d 5694	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = ( 0g ` G ) )
11		gsumval.z	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2289	. . . . . 6 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( 0g ` w ) = .0. )
13	12	eqeq2d 2250	. . . . 5 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( x = ( 0g ` w ) <-> x = .0. ) )
14	8, 13	anbi12d 477	. . . 4 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( ( dom g = (/) /\ x = ( 0g ` w ) ) <-> ( A = (/) /\ x = .0. ) ) )
15	7	eqeq1d 2247	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( dom g = ( m ... n ) <-> A = ( m ... n ) ) )
16		eqidd 2239	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> m = m )
17	9	fveq2d 5694	. . . . . . . . . . 11 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = ( +g ` G ) )
18		gsumval.p	. . . . . . . . . . 11 |- .+ = ( +g ` G )
19	17, 18	eqtr4di 2289	. . . . . . . . . 10 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( +g ` w ) = .+ )
20	16, 19, 3	seqeq123d 10871	. . . . . . . . 9 |- ( ( ph /\ ( w = G /\ g = F ) ) -> seq m ( ( +g ` w ) , g ) = seq m ( .+ , F ) )
21	20	fveq1d 5692	. . . . . . . 8 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( seq m ( ( +g ` w ) , g ) ` n ) = ( seq m ( .+ , F ) ` n ) )
22	21	eqeq2d 2250	. . . . . . 7 |- ( ( ph /\ ( w = G /\ g = F ) ) -> ( x = ( seq m ( ( +g ` w ) , g ) ` n ) <-> x = ( seq m ( .+ , F ) ` n ) ) )
23	15, 22	anbi12d 477	. . . . . 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 2551	. . . . 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 1878	. . . 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 805	. . 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 5355	. 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 2835	. 2 |- ( ph -> G e. _V )
30		gsumvalx.f	. . 3 |- ( ph -> F e. X )
31	30	elexd 2835	. 2 |- ( ph -> F e. _V )
32		unab 3498	. . . 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 3711	. . . . . . 7 |- { .0. } = { x | x = .0. }
34		fn0g 13672	. . . . . . . . . 10 |- 0g Fn _V
35		funfvex 5707	. . . . . . . . . . 11 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
36	35	funfni 5478	. . . . . . . . . 10 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
37	34, 29, 36	sylancr 418	. . . . . . . . 9 |- ( ph -> ( 0g ` G ) e. _V )
38	11, 37	eqeltrid 2325	. . . . . . . 8 |- ( ph -> .0. e. _V )
39		snexg 4316	. . . . . . . 8 |- ( .0. e. _V -> { .0. } e. _V )
40	38, 39	syl 14	. . . . . . 7 |- ( ph -> { .0. } e. _V )
41	33, 40	eqeltrrid 2326	. . . . . 6 |- ( ph -> { x | x = .0. } e. _V )
42		simpr 110	. . . . . . . 8 |- ( ( A = (/) /\ x = .0. ) -> x = .0. )
43	42	ss2abi 3320	. . . . . . 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 4268	. . . . 5 |- ( ph -> { x | ( A = (/) /\ x = .0. ) } e. _V )
46		zex 9632	. . . . . . 7 |- ZZ e. _V
47	46, 46	ab2rexex 6354	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) } e. _V
48		df-rex 2534	. . . . . . . . . . . 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 9905	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
50		eluzelz 9910	. . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . 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 1653	. . . . . . . . . . . 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 1962	. . . . . . . . . . 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 2534	. . . . . . . . . . 11 |- ( E. n e. ZZ x = ( seq m ( .+ , F ) ` n ) <-> E. n ( n e. ZZ /\ x = ( seq m ( .+ , F ) ` n ) ) )
61	60	anbi2i 461	. . . . . . . . . 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 1653	. . . . . . . 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 2534	. . . . . . . 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 3320	. . . . . 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 4266	. . . . 5 |- { x | E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) } e. _V
68		unexg 4584	. . . . 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 417	. . . 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 2326	. . 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 5351	. . 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 6206	1 |- ( ph -> ( G gzsumgz 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 720   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   _Vcvv 2821   u. cun 3218   C_ wss 3220   (/)c0 3520   {csn 3705   dom cdm 4769   iotacio 5330   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  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-ral 2533  df-rex 2534  df-reu 2535  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-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-recs 6566  df-frec 6652  df-neg 8490  df-inn 9284  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gzsumval  13687