Theorem clim2prod 12180

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 2231	. 2 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2prod.1	. . . . 5 |- Z = ( ZZ>= ` M )
3		uzssz 9837	. . . . 5 |- ( ZZ>= ` M ) C_ ZZ
4	2, 3	eqsstri 3260	. . . 4 |- Z C_ ZZ
5		clim2prod.2	. . . 4 |- ( ph -> N e. Z )
6	4, 5	sselid 3226	. . 3 |- ( ph -> N e. ZZ )
7	6	peano2zd 9666	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2prod.4	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> A )
9	5, 2	eleqtrdi 2324	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
10		eluzel2 9821	. . . . 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 12179	. . 3 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 5	ffvelcdmd 5791	. 2 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		seqex 10774	. . 3 |- seq M ( x. , F ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq M ( x. , F ) e. _V )
17		peano2uz 9878	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
18		uzss 9838	. . . . . . . 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 3277	. . . . . 6 |- ( ph -> ( ZZ>= ` ( N + 1 ) ) C_ Z )
21	20	sselda 3228	. . . . 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 12179	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
24	23	ffvelcdmda 5790	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` k ) e. CC )
25		fveq2 5648	. . . . . 6 |- ( x = ( N + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( N + 1 ) ) )
26		fveq2 5648	. . . . . . 7 |- ( x = ( N + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) )
27	26	oveq2d 6044	. . . . . 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 2246	. . . . 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 5648	. . . . . 6 |- ( x = n -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` n ) )
31		fveq2 5648	. . . . . . 7 |- ( x = n -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` n ) )
32	31	oveq2d 6044	. . . . . 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 2246	. . . . 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 5648	. . . . . 6 |- ( x = ( n + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
36		fveq2 5648	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) )
37	36	oveq2d 6044	. . . . . 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 2246	. . . . 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 5648	. . . . . 6 |- ( x = k -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` k ) )
41		fveq2 5648	. . . . . . 7 |- ( x = k -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` k ) )
42	41	oveq2d 6044	. . . . . 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 2246	. . . . 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 2298	. . . . . . . 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 8219	. . . . . . . 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 10790	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
50	7, 22, 48	seq3-1 10787	. . . . . . 7 |- ( ph -> ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) = ( F ` ( N + 1 ) ) )
51	50	oveq2d 6044	. . . . . 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 2267	. . . . 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 3228	. . . . . . . . . 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 10790	. . . . . . . . 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 6035	. . . . . . . . 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 5790	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` n ) e. CC )
63		peano2uz 9878	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. ( ZZ>= ` M ) )
64	63, 2	eleqtrrdi 2325	. . . . . . . . . . . . 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 2606	. . . . . . . . . . . . 13 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
67		fveq2 5648	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
68	67	eleq1d 2300	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
69	68	rspcv 2907	. . . . . . . . . . . . 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 8262	. . . . . . . . . 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 10790	. . . . . . . . . . 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 6044	. . . . . . . . . 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 2267	. . . . . . . 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 2268	. . . . . . 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 9883	. . 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 11971	1 |- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   C_ wss 3201   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   x. cmul 8097   ZZcz 9540   ZZ>=cuz 9816   seqcseq 10772   ~~> cli 11918

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919

This theorem is referenced by:  ntrivcvgap  12189