Theorem rlimeq 14300

Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)

Hypotheses

Ref	Expression
rlimeq.1	|- ( ( ph /\ x e. A ) -> B e. CC )
rlimeq.2	|- ( ( ph /\ x e. A ) -> C e. CC )
rlimeq.3	|- ( ph -> D e. RR )
rlimeq.4	|- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> B = C )

Assertion

Ref	Expression
rlimeq	|- ( ph -> ( ( x e. A |-> B ) ~~> r E <-> ( x e. A |-> C ) ~~> r E ) )

Distinct variable groups:   x, A   x, D   ph, x

Allowed substitution hints:   B( x)   C( x)   E( x)

Proof of Theorem rlimeq

Step	Hyp	Ref	Expression
1		rlimss 14233	. . 3 |- ( ( x e. A |-> B ) ~~> r E -> dom ( x e. A |-> B ) C_ RR )
2		eqid 2622	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
3		rlimeq.1	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. CC )
4	2, 3	dmmptd 6024	. . . 4 |- ( ph -> dom ( x e. A |-> B ) = A )
5	4	sseq1d 3632	. . 3 |- ( ph -> ( dom ( x e. A |-> B ) C_ RR <-> A C_ RR ) )
6	1, 5	syl5ib 234	. 2 |- ( ph -> ( ( x e. A |-> B ) ~~> r E -> A C_ RR ) )
7		rlimss 14233	. . 3 |- ( ( x e. A |-> C ) ~~> r E -> dom ( x e. A |-> C ) C_ RR )
8		eqid 2622	. . . . 5 |- ( x e. A |-> C ) = ( x e. A |-> C )
9		rlimeq.2	. . . . 5 |- ( ( ph /\ x e. A ) -> C e. CC )
10	8, 9	dmmptd 6024	. . . 4 |- ( ph -> dom ( x e. A |-> C ) = A )
11	10	sseq1d 3632	. . 3 |- ( ph -> ( dom ( x e. A |-> C ) C_ RR <-> A C_ RR ) )
12	7, 11	syl5ib 234	. 2 |- ( ph -> ( ( x e. A |-> C ) ~~> r E -> A C_ RR ) )
13		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> x e. ( A i^i ( D [,) +oo ) ) )
14		elin 3796	. . . . . . . . . . . . . 14 |- ( x e. ( A i^i ( D [,) +oo ) ) <-> ( x e. A /\ x e. ( D [,) +oo ) ) )
15	13, 14	sylib 208	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> ( x e. A /\ x e. ( D [,) +oo ) ) )
16	15	simpld 475	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> x e. A )
17	15	simprd 479	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> x e. ( D [,) +oo ) )
18		rlimeq.3	. . . . . . . . . . . . . . . 16 |- ( ph -> D e. RR )
19		elicopnf 12269	. . . . . . . . . . . . . . . 16 |- ( D e. RR -> ( x e. ( D [,) +oo ) <-> ( x e. RR /\ D <_ x ) ) )
20	18, 19	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( x e. ( D [,) +oo ) <-> ( x e. RR /\ D <_ x ) ) )
21	20	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( D [,) +oo ) ) -> ( x e. RR /\ D <_ x ) )
22	17, 21	syldan 487	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> ( x e. RR /\ D <_ x ) )
23	22	simprd 479	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> D <_ x )
24	16, 23	jca 554	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> ( x e. A /\ D <_ x ) )
25		rlimeq.4	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> B = C )
26	24, 25	syldan 487	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A i^i ( D [,) +oo ) ) ) -> B = C )
27	26	mpteq2dva 4744	. . . . . . . . 9 |- ( ph -> ( x e. ( A i^i ( D [,) +oo ) ) |-> B ) = ( x e. ( A i^i ( D [,) +oo ) ) |-> C ) )
28		inss1 3833	. . . . . . . . . 10 |- ( A i^i ( D [,) +oo ) ) C_ A
29		resmpt 5449	. . . . . . . . . 10 |- ( ( A i^i ( D [,) +oo ) ) C_ A -> ( ( x e. A |-> B ) |` ( A i^i ( D [,) +oo ) ) ) = ( x e. ( A i^i ( D [,) +oo ) ) |-> B ) )
30	28, 29	ax-mp 5	. . . . . . . . 9 |- ( ( x e. A |-> B ) |` ( A i^i ( D [,) +oo ) ) ) = ( x e. ( A i^i ( D [,) +oo ) ) |-> B )
31		resmpt 5449	. . . . . . . . . 10 |- ( ( A i^i ( D [,) +oo ) ) C_ A -> ( ( x e. A |-> C ) |` ( A i^i ( D [,) +oo ) ) ) = ( x e. ( A i^i ( D [,) +oo ) ) |-> C ) )
32	28, 31	ax-mp 5	. . . . . . . . 9 |- ( ( x e. A |-> C ) |` ( A i^i ( D [,) +oo ) ) ) = ( x e. ( A i^i ( D [,) +oo ) ) |-> C )
33	27, 30, 32	3eqtr4g 2681	. . . . . . . 8 |- ( ph -> ( ( x e. A |-> B ) |` ( A i^i ( D [,) +oo ) ) ) = ( ( x e. A |-> C ) |` ( A i^i ( D [,) +oo ) ) ) )
34		resres 5409	. . . . . . . 8 |- ( ( ( x e. A |-> B ) |` A ) |` ( D [,) +oo ) ) = ( ( x e. A |-> B ) |` ( A i^i ( D [,) +oo ) ) )
35		resres 5409	. . . . . . . 8 |- ( ( ( x e. A |-> C ) |` A ) |` ( D [,) +oo ) ) = ( ( x e. A |-> C ) |` ( A i^i ( D [,) +oo ) ) )
36	33, 34, 35	3eqtr4g 2681	. . . . . . 7 |- ( ph -> ( ( ( x e. A |-> B ) |` A ) |` ( D [,) +oo ) ) = ( ( ( x e. A |-> C ) |` A ) |` ( D [,) +oo ) ) )
37		ssid 3624	. . . . . . . 8 |- A C_ A
38		resmpt 5449	. . . . . . . 8 |- ( A C_ A -> ( ( x e. A |-> B ) |` A ) = ( x e. A |-> B ) )
39		reseq1 5390	. . . . . . . 8 |- ( ( ( x e. A |-> B ) |` A ) = ( x e. A |-> B ) -> ( ( ( x e. A |-> B ) |` A ) |` ( D [,) +oo ) ) = ( ( x e. A |-> B ) |` ( D [,) +oo ) ) )
40	37, 38, 39	mp2b 10	. . . . . . 7 |- ( ( ( x e. A |-> B ) |` A ) |` ( D [,) +oo ) ) = ( ( x e. A |-> B ) |` ( D [,) +oo ) )
41		resmpt 5449	. . . . . . . 8 |- ( A C_ A -> ( ( x e. A |-> C ) |` A ) = ( x e. A |-> C ) )
42		reseq1 5390	. . . . . . . 8 |- ( ( ( x e. A |-> C ) |` A ) = ( x e. A |-> C ) -> ( ( ( x e. A |-> C ) |` A ) |` ( D [,) +oo ) ) = ( ( x e. A |-> C ) |` ( D [,) +oo ) ) )
43	37, 41, 42	mp2b 10	. . . . . . 7 |- ( ( ( x e. A |-> C ) |` A ) |` ( D [,) +oo ) ) = ( ( x e. A |-> C ) |` ( D [,) +oo ) )
44	36, 40, 43	3eqtr3g 2679	. . . . . 6 |- ( ph -> ( ( x e. A |-> B ) |` ( D [,) +oo ) ) = ( ( x e. A |-> C ) |` ( D [,) +oo ) ) )
45	44	breq1d 4663	. . . . 5 |- ( ph -> ( ( ( x e. A |-> B ) |` ( D [,) +oo ) ) ~~> r E <-> ( ( x e. A |-> C ) |` ( D [,) +oo ) ) ~~> r E ) )
46	45	adantr 481	. . . 4 |- ( ( ph /\ A C_ RR ) -> ( ( ( x e. A |-> B ) |` ( D [,) +oo ) ) ~~> r E <-> ( ( x e. A |-> C ) |` ( D [,) +oo ) ) ~~> r E ) )
47	3, 2	fmptd 6385	. . . . . 6 |- ( ph -> ( x e. A |-> B ) : A --> CC )
48	47	adantr 481	. . . . 5 |- ( ( ph /\ A C_ RR ) -> ( x e. A |-> B ) : A --> CC )
49		simpr 477	. . . . 5 |- ( ( ph /\ A C_ RR ) -> A C_ RR )
50	18	adantr 481	. . . . 5 |- ( ( ph /\ A C_ RR ) -> D e. RR )
51	48, 49, 50	rlimresb 14296	. . . 4 |- ( ( ph /\ A C_ RR ) -> ( ( x e. A |-> B ) ~~> r E <-> ( ( x e. A |-> B ) |` ( D [,) +oo ) ) ~~> r E ) )
52	9, 8	fmptd 6385	. . . . . 6 |- ( ph -> ( x e. A |-> C ) : A --> CC )
53	52	adantr 481	. . . . 5 |- ( ( ph /\ A C_ RR ) -> ( x e. A |-> C ) : A --> CC )
54	53, 49, 50	rlimresb 14296	. . . 4 |- ( ( ph /\ A C_ RR ) -> ( ( x e. A |-> C ) ~~> r E <-> ( ( x e. A |-> C ) |` ( D [,) +oo ) ) ~~> r E ) )
55	46, 51, 54	3bitr4d 300	. . 3 |- ( ( ph /\ A C_ RR ) -> ( ( x e. A |-> B ) ~~> r E <-> ( x e. A |-> C ) ~~> r E ) )
56	55	ex 450	. 2 |- ( ph -> ( A C_ RR -> ( ( x e. A |-> B ) ~~> r E <-> ( x e. A |-> C ) ~~> r E ) ) )
57	6, 12, 56	pm5.21ndd 369	1 |- ( ph -> ( ( x e. A |-> B ) ~~> r E <-> ( x e. A |-> C ) ~~> r E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   -->wf 5884  (class class class)co 6650   CCcc 9934   RRcr 9935   +oocpnf 10071   <_ cle 10075   [,)cico 12177   ~~> r crli 14216

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181  df-rlim 14220

This theorem is referenced by: (None)