Theorem clim2prod 11704

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 2196	. 2 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2prod.1	. . . . 5 |- Z = ( ZZ>= ` M )
3		uzssz 9621	. . . . 5 |- ( ZZ>= ` M ) C_ ZZ
4	2, 3	eqsstri 3215	. . . 4 |- Z C_ ZZ
5		clim2prod.2	. . . 4 |- ( ph -> N e. Z )
6	4, 5	sselid 3181	. . 3 |- ( ph -> N e. ZZ )
7	6	peano2zd 9451	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2prod.4	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> A )
9	5, 2	eleqtrdi 2289	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
10		eluzel2 9606	. . . . 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 11703	. . 3 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 5	ffvelcdmd 5698	. 2 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		seqex 10541	. . 3 |- seq M ( x. , F ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq M ( x. , F ) e. _V )
17		peano2uz 9657	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
18		uzss 9622	. . . . . . . 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 3232	. . . . . 6 |- ( ph -> ( ZZ>= ` ( N + 1 ) ) C_ Z )
21	20	sselda 3183	. . . . 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 11703	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
24	23	ffvelcdmda 5697	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` k ) e. CC )
25		fveq2 5558	. . . . . 6 |- ( x = ( N + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( N + 1 ) ) )
26		fveq2 5558	. . . . . . 7 |- ( x = ( N + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) )
27	26	oveq2d 5938	. . . . . 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 2211	. . . . 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 5558	. . . . . 6 |- ( x = n -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` n ) )
31		fveq2 5558	. . . . . . 7 |- ( x = n -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` n ) )
32	31	oveq2d 5938	. . . . . 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 2211	. . . . 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 5558	. . . . . 6 |- ( x = ( n + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
36		fveq2 5558	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) )
37	36	oveq2d 5938	. . . . . 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 2211	. . . . 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 5558	. . . . . 6 |- ( x = k -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` k ) )
41		fveq2 5558	. . . . . . 7 |- ( x = k -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` k ) )
42	41	oveq2d 5938	. . . . . 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 2211	. . . . 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 2263	. . . . . . . 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 8006	. . . . . . . 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 10557	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
50	7, 22, 48	seq3-1 10554	. . . . . . 7 |- ( ph -> ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) = ( F ` ( N + 1 ) ) )
51	50	oveq2d 5938	. . . . . 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 2232	. . . . 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 3183	. . . . . . . . . 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 10557	. . . . . . . . 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 5929	. . . . . . . . 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 5697	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` n ) e. CC )
63		peano2uz 9657	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. ( ZZ>= ` M ) )
64	63, 2	eleqtrrdi 2290	. . . . . . . . . . . . 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 2570	. . . . . . . . . . . . 13 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
67		fveq2 5558	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
68	67	eleq1d 2265	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
69	68	rspcv 2864	. . . . . . . . . . . . 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 8050	. . . . . . . . . 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 10557	. . . . . . . . . . 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 5938	. . . . . . . . . 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 2232	. . . . . . . 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 2233	. . . . . . 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 9662	. . 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 11496	1 |- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   C_ wss 3157   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   1c1 7880   + caddc 7882   x. cmul 7884   ZZcz 9326   ZZ>=cuz 9601   seqcseq 10539   ~~> cli 11443

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444

This theorem is referenced by:  ntrivcvgap  11713