Theorem clim2prod 14620

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

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2prod.1	. . . . 5 |- Z = ( ZZ>= ` M )
3		uzssz 11707	. . . . 5 |- ( ZZ>= ` M ) C_ ZZ
4	2, 3	eqsstri 3635	. . . 4 |- Z C_ ZZ
5		clim2prod.2	. . . 4 |- ( ph -> N e. Z )
6	4, 5	sseldi 3601	. . 3 |- ( ph -> N e. ZZ )
7	6	peano2zd 11485	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2prod.4	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> A )
9	5, 2	syl6eleq 2711	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
10		eluzel2 11692	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
11	9, 10	syl 17	. . . 4 |- ( ph -> M e. ZZ )
12		clim2prod.3	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	2, 11, 12	prodf 14619	. . 3 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 5	ffvelrnd 6360	. 2 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		seqex 12803	. . 3 |- seq M ( x. , F ) e. _V
16	15	a1i 11	. 2 |- ( ph -> seq M ( x. , F ) e. _V )
17		peano2uz 11741	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
18		uzss 11708	. . . . . . . 8 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( ZZ>= ` ( N + 1 ) ) C_ ( ZZ>= ` M ) )
19	9, 17, 18	3syl 18	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( N + 1 ) ) C_ ( ZZ>= ` M ) )
20	19, 2	syl6sseqr 3652	. . . . . 6 |- ( ph -> ( ZZ>= ` ( N + 1 ) ) C_ Z )
21	20	sselda 3603	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
22	21, 12	syldan 487	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
23	1, 7, 22	prodf 14619	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
24	23	ffvelrnda 6359	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` k ) e. CC )
25		fveq2 6191	. . . . . 6 |- ( x = ( N + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( N + 1 ) ) )
26		fveq2 6191	. . . . . . 7 |- ( x = ( N + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) )
27	26	oveq2d 6666	. . . . . 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 2637	. . . . 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 330	. . . 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 6191	. . . . . 6 |- ( x = n -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` n ) )
31		fveq2 6191	. . . . . . 7 |- ( x = n -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` n ) )
32	31	oveq2d 6666	. . . . . 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 2637	. . . . 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 330	. . . 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 6191	. . . . . 6 |- ( x = ( n + 1 ) -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
36		fveq2 6191	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) )
37	36	oveq2d 6666	. . . . . 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 2637	. . . . 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 330	. . . 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 6191	. . . . . 6 |- ( x = k -> ( seq M ( x. , F ) ` x ) = ( seq M ( x. , F ) ` k ) )
41		fveq2 6191	. . . . . . 7 |- ( x = k -> ( seq ( N + 1 ) ( x. , F ) ` x ) = ( seq ( N + 1 ) ( x. , F ) ` k ) )
42	41	oveq2d 6666	. . . . . 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 2637	. . . . 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 330	. . . 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	9	adantr 481	. . . . . . 7 |- ( ( ph /\ ( N + 1 ) e. ZZ ) -> N e. ( ZZ>= ` M ) )
46		seqp1 12816	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
47	45, 46	syl 17	. . . . . 6 |- ( ( ph /\ ( N + 1 ) e. ZZ ) -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
48		seq1 12814	. . . . . . . 8 |- ( ( N + 1 ) e. ZZ -> ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) = ( F ` ( N + 1 ) ) )
49	48	adantl 482	. . . . . . 7 |- ( ( ph /\ ( N + 1 ) e. ZZ ) -> ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) = ( F ` ( N + 1 ) ) )
50	49	oveq2d 6666	. . . . . 6 |- ( ( ph /\ ( N + 1 ) e. ZZ ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) = ( ( seq M ( x. , F ) ` N ) x. ( F ` ( N + 1 ) ) ) )
51	47, 50	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( N + 1 ) e. ZZ ) -> ( seq M ( x. , F ) ` ( N + 1 ) ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` ( N + 1 ) ) ) )
52	51	expcom 451	. . . 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 ) ) ) ) )
53	19	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> n e. ( ZZ>= ` M ) )
54		seqp1 12816	. . . . . . . . . 10 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
55	53, 54	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
56	55	adantr 481	. . . . . . . 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 ) ) ) )
57		oveq1 6657	. . . . . . . . 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 ) ) ) )
58	57	adantl 482	. . . . . . . 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 ) ) ) )
59	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
60	23	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` n ) e. CC )
61		peano2uz 11741	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. ( ZZ>= ` M ) )
62	61, 2	syl6eleqr 2712	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` M ) -> ( n + 1 ) e. Z )
63	53, 62	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( n + 1 ) e. Z )
64	12	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
65		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
66	65	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
67	66	rspcv 3305	. . . . . . . . . . . . 13 |- ( ( n + 1 ) e. Z -> ( A. k e. Z ( F ` k ) e. CC -> ( F ` ( n + 1 ) ) e. CC ) )
68	64, 67	mpan9 486	. . . . . . . . . . . 12 |- ( ( ph /\ ( n + 1 ) e. Z ) -> ( F ` ( n + 1 ) ) e. CC )
69	63, 68	syldan 487	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` ( n + 1 ) ) e. CC )
70	59, 60, 69	mulassd 10063	. . . . . . . . . 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 ) ) ) ) )
71	70	adantr 481	. . . . . . . . 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 ) ) ) ) )
72		seqp1 12816	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` ( N + 1 ) ) -> ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) = ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
73	72	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` ( n + 1 ) ) = ( ( seq ( N + 1 ) ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
74	73	oveq2d 6666	. . . . . . . . . 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 ) ) ) ) )
75	74	adantr 481	. . . . . . . . 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 ) ) ) ) )
76	71, 75	eqtr4d 2659	. . . . . . . 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 ) ) ) )
77	56, 58, 76	3eqtrd 2660	. . . . . . 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 ) ) ) )
78	77	exp31 630	. . . . . 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 ) ) ) ) ) )
79	78	com12 32	. . . . 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 ) ) ) ) ) )
80	79	a2d 29	. . . 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 ) ) ) ) ) )
81	29, 34, 39, 44, 52, 80	uzind4 11746	. . 3 |- ( k e. ( ZZ>= ` ( N + 1 ) ) -> ( ph -> ( seq M ( x. , F ) ` k ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) ) )
82	81	impcom 446	. 2 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` k ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` k ) ) )
83	1, 7, 8, 14, 16, 24, 82	climmulc2 14367	1 |- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  ntrivcvg  14629