Theorem sumeq2 8245

Description: Equality theorem for sum.

Assertion

Ref	Expression
sumeq2	|- (A.k e. A B = C -> sum_k e. A B = sum_k e. A C)

Distinct variable group:   A,k

Proof of Theorem sumeq2

Step	Hyp	Ref	Expression
1		hbra1 2147	. . . . . . . . . . . . . . . 16 |- (A.k e. A B = C -> A.kA.k e. A B = C)
2		ax-17 1317	. . . . . . . . . . . . . . . 16 |- (A.k e. A B = C -> A.yA.k e. A B = C)
3		ra4 2155	. . . . . . . . . . . . . . . . . . 19 |- (A.k e. A B = C -> (k e. A -> B = C))
4	3	imp 377	. . . . . . . . . . . . . . . . . 18 |- ((A.k e. A B = C /\ k e. A) -> B = C)
5	4	eqeq2d 1895	. . . . . . . . . . . . . . . . 17 |- ((A.k e. A B = C /\ k e. A) -> (y = B <-> y = C))
6	5	pm5.32da 711	. . . . . . . . . . . . . . . 16 |- (A.k e. A B = C -> ((k e. A /\ y = B) <-> (k e. A /\ y = C)))
7	1, 2, 6	opabbid 3399	. . . . . . . . . . . . . . 15 |- (A.k e. A B = C -> {<.k, y>. | (k e. A /\ y = B)} = {<.k, y>. | (k e. A /\ y = C)})
8		resopab 4252	. . . . . . . . . . . . . . 15 |- ({<.k, y>. | y = B} |` A) = {<.k, y>. | (k e. A /\ y = B)}
9		resopab 4252	. . . . . . . . . . . . . . 15 |- ({<.k, y>. | y = C} |` A) = {<.k, y>. | (k e. A /\ y = C)}
10	7, 8, 9	3eqtr4g 1953	. . . . . . . . . . . . . 14 |- (A.k e. A B = C -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = C} |` A))
11	10	adantr 425	. . . . . . . . . . . . 13 |- ((A.k e. A B = C /\ A = (m...n)) -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = C} |` A))
12		reseq2 4219	. . . . . . . . . . . . . 14 |- (A = (m...n) -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = B} |` (m...n)))
13	12	adantl 424	. . . . . . . . . . . . 13 |- ((A.k e. A B = C /\ A = (m...n)) -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = B} |` (m...n)))
14		reseq2 4219	. . . . . . . . . . . . . 14 |- (A = (m...n) -> ({<.k, y>. | y = C} |` A) = ({<.k, y>. | y = C} |` (m...n)))
15	14	adantl 424	. . . . . . . . . . . . 13 |- ((A.k e. A B = C /\ A = (m...n)) -> ({<.k, y>. | y = C} |` A) = ({<.k, y>. | y = C} |` (m...n)))
16	11, 13, 15	3eqtr3d 1934	. . . . . . . . . . . 12 |- ((A.k e. A B = C /\ A = (m...n)) -> ({<.k, y>. | y = B} |` (m...n)) = ({<.k, y>. | y = C} |` (m...n)))
17	16	opreq2d 4898	. . . . . . . . . . 11 |- ((A.k e. A B = C /\ A = (m...n)) -> (<.m, + >. seq ({<.k, y>. | y = B} |` (m...n))) = (<.m, + >. seq ({<.k, y>. | y = C} |` (m...n))))
18	17	fveq1d 4683	. . . . . . . . . 10 |- ((A.k e. A B = C /\ A = (m...n)) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` (m...n)))` n) = ((<.m, + >. seq ({<.k, y>. | y = C} |` (m...n)))` n))
19	18	adantlr 429	. . . . . . . . 9 |- (((A.k e. A B = C /\ n e. (ZZ>=` m)) /\ A = (m...n)) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` (m...n)))` n) = ((<.m, + >. seq ({<.k, y>. | y = C} |` (m...n)))` n))
20		elfzelz 7652	. . . . . . . . . . . . . 14 |- (x e. (m...n) -> x e. ZZ)
21		fvres 4691	. . . . . . . . . . . . . 14 |- (x e. ZZ -> (({<.k, y>. | y = B} |` ZZ)` x) = ({<.k, y>. | y = B}` x))
22	20, 21	syl 12	. . . . . . . . . . . . 13 |- (x e. (m...n) -> (({<.k, y>. | y = B} |` ZZ)` x) = ({<.k, y>. | y = B}` x))
23		fvres 4691	. . . . . . . . . . . . 13 |- (x e. (m...n) -> (({<.k, y>. | y = B} |` (m...n))` x) = ({<.k, y>. | y = B}` x))
24	22, 23	eqtr4d 1928	. . . . . . . . . . . 12 |- (x e. (m...n) -> (({<.k, y>. | y = B} |` ZZ)` x) = (({<.k, y>. | y = B} |` (m...n))` x))
25	24	rgen 2159	. . . . . . . . . . 11 |- A.x e. (m...n)(({<.k, y>. | y = B} |` ZZ)` x) = (({<.k, y>. | y = B} |` (m...n))` x)
26		addex 6470	. . . . . . . . . . . 12 |- + e. _V
27		funopabeq 4456	. . . . . . . . . . . . 13 |- Fun {<.k, y>. | y = B}
28		zex 7353	. . . . . . . . . . . . 13 |- ZZ e. _V
29		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun {<.k, y>. | y = B} /\ ZZ e. _V) -> ({<.k, y>. | y = B} |` ZZ) e. _V)
30	27, 28, 29	mp2an 761	. . . . . . . . . . . 12 |- ({<.k, y>. | y = B} |` ZZ) e. _V
31		oprex 4907	. . . . . . . . . . . . 13 |- (m...n) e. _V
32		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun {<.k, y>. | y = B} /\ (m...n) e. _V) -> ({<.k, y>. | y = B} |` (m...n)) e. _V)
33	27, 31, 32	mp2an 761	. . . . . . . . . . . 12 |- ({<.k, y>. | y = B} |` (m...n)) e. _V
34	26, 30, 33	seqzfveq 7797	. . . . . . . . . . 11 |- ((n e. (ZZ>=` m) /\ A.x e. (m...n)(({<.k, y>. | y = B} |` ZZ)` x) = (({<.k, y>. | y = B} |` (m...n))` x)) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = B} |` (m...n)))` n))
35	25, 34	mpan2 760	. . . . . . . . . 10 |- (n e. (ZZ>=` m) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = B} |` (m...n)))` n))
36	35	ad2antlr 441	. . . . . . . . 9 |- (((A.k e. A B = C /\ n e. (ZZ>=` m)) /\ A = (m...n)) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = B} |` (m...n)))` n))
37		fvres 4691	. . . . . . . . . . . . . 14 |- (x e. ZZ -> (({<.k, y>. | y = C} |` ZZ)` x) = ({<.k, y>. | y = C}` x))
38	20, 37	syl 12	. . . . . . . . . . . . 13 |- (x e. (m...n) -> (({<.k, y>. | y = C} |` ZZ)` x) = ({<.k, y>. | y = C}` x))
39		fvres 4691	. . . . . . . . . . . . 13 |- (x e. (m...n) -> (({<.k, y>. | y = C} |` (m...n))` x) = ({<.k, y>. | y = C}` x))
40	38, 39	eqtr4d 1928	. . . . . . . . . . . 12 |- (x e. (m...n) -> (({<.k, y>. | y = C} |` ZZ)` x) = (({<.k, y>. | y = C} |` (m...n))` x))
41	40	rgen 2159	. . . . . . . . . . 11 |- A.x e. (m...n)(({<.k, y>. | y = C} |` ZZ)` x) = (({<.k, y>. | y = C} |` (m...n))` x)
42		funopabeq 4456	. . . . . . . . . . . . 13 |- Fun {<.k, y>. | y = C}
43		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun {<.k, y>. | y = C} /\ ZZ e. _V) -> ({<.k, y>. | y = C} |` ZZ) e. _V)
44	42, 28, 43	mp2an 761	. . . . . . . . . . . 12 |- ({<.k, y>. | y = C} |` ZZ) e. _V
45		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun {<.k, y>. | y = C} /\ (m...n) e. _V) -> ({<.k, y>. | y = C} |` (m...n)) e. _V)
46	42, 31, 45	mp2an 761	. . . . . . . . . . . 12 |- ({<.k, y>. | y = C} |` (m...n)) e. _V
47	26, 44, 46	seqzfveq 7797	. . . . . . . . . . 11 |- ((n e. (ZZ>=` m) /\ A.x e. (m...n)(({<.k, y>. | y = C} |` ZZ)` x) = (({<.k, y>. | y = C} |` (m...n))` x)) -> ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = C} |` (m...n)))` n))
48	41, 47	mpan2 760	. . . . . . . . . 10 |- (n e. (ZZ>=` m) -> ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = C} |` (m...n)))` n))
49	48	ad2antlr 441	. . . . . . . . 9 |- (((A.k e. A B = C /\ n e. (ZZ>=` m)) /\ A = (m...n)) -> ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = C} |` (m...n)))` n))
50	19, 36, 49	3eqtr4d 1937	. . . . . . . 8 |- (((A.k e. A B = C /\ n e. (ZZ>=` m)) /\ A = (m...n)) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n) = ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))
51	50	eleq2d 1964	. . . . . . 7 |- (((A.k e. A B = C /\ n e. (ZZ>=` m)) /\ A = (m...n)) -> (x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n) <-> x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n)))
52	51	pm5.32da 711	. . . . . 6 |- ((A.k e. A B = C /\ n e. (ZZ>=` m)) -> ((A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n)) <-> (A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))))
53	52	rexbidva 2120	. . . . 5 |- (A.k e. A B = C -> (E.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n)) <-> E.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))))
54	53	exbidv 1657	. . . 4 |- (A.k e. A B = C -> (E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n)) <-> E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))))
55	54	abbidv 2008	. . 3 |- (A.k e. A B = C -> {x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n))} = {x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))})
56	10	adantr 425	. . . . . . . . . . . 12 |- ((A.k e. A B = C /\ A = (ZZ>=` m)) -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = C} |` A))
57		reseq2 4219	. . . . . . . . . . . . 13 |- (A = (ZZ>=` m) -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = B} |` (ZZ>=` m)))
58	57	adantl 424	. . . . . . . . . . . 12 |- ((A.k e. A B = C /\ A = (ZZ>=` m)) -> ({<.k, y>. | y = B} |` A) = ({<.k, y>. | y = B} |` (ZZ>=` m)))
59		reseq2 4219	. . . . . . . . . . . . 13 |- (A = (ZZ>=` m) -> ({<.k, y>. | y = C} |` A) = ({<.k, y>. | y = C} |` (ZZ>=` m)))
60	59	adantl 424	. . . . . . . . . . . 12 |- ((A.k e. A B = C /\ A = (ZZ>=` m)) -> ({<.k, y>. | y = C} |` A) = ({<.k, y>. | y = C} |` (ZZ>=` m)))
61	56, 58, 60	3eqtr3d 1934	. . . . . . . . . . 11 |- ((A.k e. A B = C /\ A = (ZZ>=` m)) -> ({<.k, y>. | y = B} |` (ZZ>=` m)) = ({<.k, y>. | y = C} |` (ZZ>=` m)))
62	61	opreq2d 4898	. . . . . . . . . 10 |- ((A.k e. A B = C /\ A = (ZZ>=` m)) -> (<.m, + >. seq ({<.k, y>. | y = B} |` (ZZ>=` m))) = (<.m, + >. seq ({<.k, y>. | y = C} |` (ZZ>=` m))))
63	62	adantlr 429	. . . . . . . . 9 |- (((A.k e. A B = C /\ m e. ZZ) /\ A = (ZZ>=` m)) -> (<.m, + >. seq ({<.k, y>. | y = B} |` (ZZ>=` m))) = (<.m, + >. seq ({<.k, y>. | y = C} |` (ZZ>=` m))))
64		eluzelz 7592	. . . . . . . . . . . . . 14 |- (x e. (ZZ>=` m) -> x e. ZZ)
65	64, 21	syl 12	. . . . . . . . . . . . 13 |- (x e. (ZZ>=` m) -> (({<.k, y>. | y = B} |` ZZ)` x) = ({<.k, y>. | y = B}` x))
66		fvres 4691	. . . . . . . . . . . . 13 |- (x e. (ZZ>=` m) -> (({<.k, y>. | y = B} |` (ZZ>=` m))` x) = ({<.k, y>. | y = B}` x))
67	65, 66	eqtr4d 1928	. . . . . . . . . . . 12 |- (x e. (ZZ>=` m) -> (({<.k, y>. | y = B} |` ZZ)` x) = (({<.k, y>. | y = B} |` (ZZ>=` m))` x))
68	67	rgen 2159	. . . . . . . . . . 11 |- A.x e. (ZZ>=` m)(({<.k, y>. | y = B} |` ZZ)` x) = (({<.k, y>. | y = B} |` (ZZ>=` m))` x)
69		fvex 4689	. . . . . . . . . . . . 13 |- (ZZ>=` m) e. _V
70		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun {<.k, y>. | y = B} /\ (ZZ>=` m) e. _V) -> ({<.k, y>. | y = B} |` (ZZ>=` m)) e. _V)
71	27, 69, 70	mp2an 761	. . . . . . . . . . . 12 |- ({<.k, y>. | y = B} |` (ZZ>=` m)) e. _V
72	26, 30, 71	seqzeq 7798	. . . . . . . . . . 11 |- ((m e. ZZ /\ A.x e. (ZZ>=` m)(({<.k, y>. | y = B} |` ZZ)` x) = (({<.k, y>. | y = B} |` (ZZ>=` m))` x)) -> (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = B} |` (ZZ>=` m))))
73	68, 72	mpan2 760	. . . . . . . . . 10 |- (m e. ZZ -> (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = B} |` (ZZ>=` m))))
74	73	ad2antlr 441	. . . . . . . . 9 |- (((A.k e. A B = C /\ m e. ZZ) /\ A = (ZZ>=` m)) -> (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = B} |` (ZZ>=` m))))
75	64, 37	syl 12	. . . . . . . . . . . . 13 |- (x e. (ZZ>=` m) -> (({<.k, y>. | y = C} |` ZZ)` x) = ({<.k, y>. | y = C}` x))
76		fvres 4691	. . . . . . . . . . . . 13 |- (x e. (ZZ>=` m) -> (({<.k, y>. | y = C} |` (ZZ>=` m))` x) = ({<.k, y>. | y = C}` x))
77	75, 76	eqtr4d 1928	. . . . . . . . . . . 12 |- (x e. (ZZ>=` m) -> (({<.k, y>. | y = C} |` ZZ)` x) = (({<.k, y>. | y = C} |` (ZZ>=` m))` x))
78	77	rgen 2159	. . . . . . . . . . 11 |- A.x e. (ZZ>=` m)(({<.k, y>. | y = C} |` ZZ)` x) = (({<.k, y>. | y = C} |` (ZZ>=` m))` x)
79		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun {<.k, y>. | y = C} /\ (ZZ>=` m) e. _V) -> ({<.k, y>. | y = C} |` (ZZ>=` m)) e. _V)
80	42, 69, 79	mp2an 761	. . . . . . . . . . . 12 |- ({<.k, y>. | y = C} |` (ZZ>=` m)) e. _V
81	26, 44, 80	seqzeq 7798	. . . . . . . . . . 11 |- ((m e. ZZ /\ A.x e. (ZZ>=` m)(({<.k, y>. | y = C} |` ZZ)` x) = (({<.k, y>. | y = C} |` (ZZ>=` m))` x)) -> (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = C} |` (ZZ>=` m))))
82	78, 81	mpan2 760	. . . . . . . . . 10 |- (m e. ZZ -> (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = C} |` (ZZ>=` m))))
83	82	ad2antlr 441	. . . . . . . . 9 |- (((A.k e. A B = C /\ m e. ZZ) /\ A = (ZZ>=` m)) -> (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = C} |` (ZZ>=` m))))
84	63, 74, 83	3eqtr4d 1937	. . . . . . . 8 |- (((A.k e. A B = C /\ m e. ZZ) /\ A = (ZZ>=` m)) -> (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) = (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)))
85	84	breq1d 3348	. . . . . . 7 |- (((A.k e. A B = C /\ m e. ZZ) /\ A = (ZZ>=` m)) -> ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x <-> (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x))
86	85	pm5.32da 711	. . . . . 6 |- ((A.k e. A B = C /\ m e. ZZ) -> ((A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x) <-> (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x)))
87	86	rexbidva 2120	. . . . 5 |- (A.k e. A B = C -> (E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x) <-> E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x)))
88	87	abbidv 2008	. . . 4 |- (A.k e. A B = C -> {x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x)} = {x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x)})
89	88	unieqd 3188	. . 3 |- (A.k e. A B = C -> U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x)} = U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x)})
90	55, 89	uneq12d 2756	. 2 |- (A.k e. A B = C -> ({x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n))} u. U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x)}) = ({x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))} u. U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x)}))
91		df-sum 8240	. 2 |- sum_k e. A B = ({x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n))} u. U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x)})
92		df-sum 8240	. 2 |- sum_k e. A C = ({x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = C} |` ZZ))` n))} u. U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = C} |` ZZ)) ~~> x)})
93	90, 91, 92	3eqtr4g 1953	1 |- (A.k e. A B = C -> sum_k e. A B = sum_k e. A C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591  <.cop 3046  U.cuni 3177   class class class wbr 3338  {copab 3395   |` cres 3988  Fun wfun 3992  ` cfv 3998  (class class class)co 4884   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774   ~~> cli 8234  sum_csu 8239

This theorem is referenced by:  sumeq2i 8248  sumeq2d 8251  sumeq2dv 8252  2sumeq2dv 8254  fsump1i 8266  serz0 8313  fsum00 15820

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-sum 8240

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