Theorem clim2prod 11850

Description: The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)

Hypotheses

Ref	Expression
clim2prod.1	|- Z = ( ZZ>= ` M )
clim2prod.2	|- ( ph -> N e. Z )
clim2prod.3	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
clim2prod.4	|- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> A )

Assertion

Ref	Expression
clim2prod	|- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) )

Distinct variable groups:   A, k   k, F   ph, k   k, M   k, N   k, Z

Proof of Theorem clim2prod

Dummy variables v n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2205	. 2 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2prod.1	. . . . 5 |- Z = ( ZZ>= ` M )
3		uzssz 9668	. . . . 5 |- ( ZZ>= ` M ) C_ ZZ
4	2, 3	eqsstri 3225	. . . 4 |- Z C_ ZZ
5		clim2prod.2	. . . 4 |- ( ph -> N e. Z )
6	4, 5	sselid 3191	. . 3 |- ( ph -> N e. ZZ )
7	6	peano2zd 9498	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2prod.4	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> A )
9	5, 2	eleqtrdi 2298	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
10		eluzel2 9653	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
11	9, 10	syl 14	. . . 4 |- ( ph -> M e. ZZ )
12		clim2prod.3	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	2, 11, 12	prodf 11849	. . 3 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 5	ffvelcdmd 5716	. 2 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		seqex 10594	. . 3 |- seq M ( x. , F ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq M ( x. , F ) e. _V )
17		peano2uz 9704	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
18		uzss 9669	. . . . . . . 8 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( ZZ>= ` ( N + 1 ) ) C_ ( ZZ>= ` M ) )
19	9, 17, 18	3syl 17	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( N + 1 ) ) C_ ( ZZ>= ` M ) )
20	19, 2	sseqtrrdi 3242	. . . . . 6 |- ( ph -> ( ZZ>= ` ( N + 1 ) ) C_ Z )
21	20	sselda 3193	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
22	21, 12	syldan 282	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
23	1, 7, 22	prodf 11849	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
24	23	ffvelcdmda 5715	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` k ) e. CC )
25		fveq2 5576	. . . . . 6 |- ( x = ( N + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( N + 1 ) ) )
26		fveq2 5576	. . . . . . 7 |- ( x = ( N + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) )
27	26	oveq2d 5960	. . . . . 6 |- ( x = ( N + 1 ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) )
28	25, 27	eqeq12d 2220	. . . . 5 |- ( x = ( N + 1 ) -> ( ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) <-> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) ) )
29	28	imbi2d 230	. . . 4 |- ( x = ( N + 1 ) -> ( ( ph -> ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) ) <-> ( ph -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) ) ) )
30		fveq2 5576	. . . . . 6 |- ( x = n -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` n ) )
31		fveq2 5576	. . . . . . 7 |- ( x = n -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` n ) )
32	31	oveq2d 5960	. . . . . 6 |- ( x = n -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) )
33	30, 32	eqeq12d 2220	. . . . 5 |- ( x = n -> ( ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) <-> ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) )
34	33	imbi2d 230	. . . 4 |- ( x = n -> ( ( ph -> ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) ) <-> ( ph -> ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) ) )
35		fveq2 5576	. . . . . 6 |- ( x = ( n + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
36		fveq2 5576	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) )
37	36	oveq2d 5960	. . . . . 6 |- ( x = ( n + 1 ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) )
38	35, 37	eqeq12d 2220	. . . . 5 |- ( x = ( n + 1 ) -> ( ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) <-> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) ) )
39	38	imbi2d 230	. . . 4 |- ( x = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) ) <-> ( ph -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) ) ) )
40		fveq2 5576	. . . . . 6 |- ( x = k -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` k ) )
41		fveq2 5576	. . . . . . 7 |- ( x = k -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` k ) )
42	41	oveq2d 5960	. . . . . 6 |- ( x = k -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) )
43	40, 42	eqeq12d 2220	. . . . 5 |- ( x = k -> ( ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) <-> ( seq M ( x. , F ) ` k ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) ) )
44	43	imbi2d 230	. . . 4 |- ( x = k -> ( ( ph -> ( seq M ( x. , F ) ` x ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` x ) ) ) <-> ( ph -> ( seq M ( x. , F ) ` k ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) ) ) )
45	2	eleq2i 2272	. . . . . . . 8 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
46	45, 12	sylan2br 288	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
47		mulcl 8052	. . . . . . . 8 |- ( ( k e. CC /\ v e. CC ) -> ( k x. v ) e. CC )
48	47	adantl 277	. . . . . . 7 |- ( ( ph /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
49	9, 46, 48	seq3p1 10610	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
50	7, 22, 48	seq3-1 10607	. . . . . . 7 |- ( ph -> ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) = ( F ` ( N + 1 ) ) )
51	50	oveq2d 5960	. . . . . 6 |- ( ph -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
52	49, 51	eqtr4d 2241	. . . . 5 |- ( ph -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) )
53	52	a1i 9	. . . 4 |- ( ( N + 1 ) e. ZZ -> ( ph -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) ) )
54	19	sselda 3193	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> n e. ( ZZ>= ` M ) )
55	46	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
56	47	adantl 277	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
57	54, 55, 56	seq3p1 10610	. . . . . . . . 9 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
58	57	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
59		oveq1 5951	. . . . . . . . 9 |- ( ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) x. ( F ` ( n + 1 ) ) ) )
60	59	adantl 277	. . . . . . . 8 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) x. ( F ` ( n + 1 ) ) ) )
61	14	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
62	23	ffvelcdmda 5715	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` n ) e. CC )
63		peano2uz 9704	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. ( ZZ>= ` M ) )
64	63, 2	eleqtrrdi 2299	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. Z )
65	54, 64	syl 14	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( n + 1 ) e. Z )
66	12	ralrimiva 2579	. . . . . . . . . . . . 13 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
67		fveq2 5576	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
68	67	eleq1d 2274	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
69	68	rspcv 2873	. . . . . . . . . . . . 13 |- ( ( n + 1 ) e. Z -> ( A. k e. Z ( F ` k ) e. CC -> ( F ` ( n + 1 ) ) e. CC ) )
70	66, 69	mpan9 281	. . . . . . . . . . . 12 |- ( ( ph /\ ( n + 1 ) e. Z ) -> ( F ` ( n + 1 ) ) e. CC )
71	65, 70	syldan 282	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` ( n + 1 ) ) e. CC )
72	61, 62, 71	mulassd 8096	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) x. ( F ` ( n + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
73	72	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) x. ( F ` ( n + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
74		simpr 110	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> n e. ( ZZ>= ` ( N + 1 ) ) )
75	22	adantlr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
76	74, 75, 56	seq3p1 10610	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) = ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
77	76	oveq2d 5960	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
78	77	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
79	73, 78	eqtr4d 2241	. . . . . . . 8 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) x. ( F ` ( n + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) )
80	58, 60, 79	3eqtrd 2242	. . . . . . 7 |- ( ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) )
81	80	exp31 364	. . . . . 6 |- ( ph -> ( n e. ( ZZ>= ` ( N + 1 ) ) -> ( ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) ) ) )
82	81	com12 30	. . . . 5 |- ( n e. ( ZZ>= ` ( N + 1 ) ) -> ( ph -> ( ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) ) ) )
83	82	a2d 26	. . . 4 |- ( n e. ( ZZ>= ` ( N + 1 ) ) -> ( ( ph -> ( seq M ( x. , F ) ` n ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` n ) ) ) -> ( ph -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) ) ) ) )
84	29, 34, 39, 44, 53, 83	uzind4 9709	. . 3 |- ( k e. ( ZZ>= ` ( N + 1 ) ) -> ( ph -> ( seq M ( x. , F ) ` k ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) ) )
85	84	impcom 125	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` k ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) )
86	1, 7, 8, 14, 16, 24, 85	climmulc2 11642	1 |- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   1c1 7926   + caddc 7928   x. cmul 7930   ZZcz 9372   ZZ>=cuz 9648   seqcseq 10592   ~~> cli 11589

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590

This theorem is referenced by:  ntrivcvgap  11859