Theorem lmcn2 15074

Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)

Hypotheses

Ref	Expression
txlm.z	|- Z = ( ZZ>= ` M )
txlm.m	|- ( ph -> M e. ZZ )
txlm.j	|- ( ph -> J e. (TopOn ` X ) )
txlm.k	|- ( ph -> K e. (TopOn ` Y ) )
txlm.f	|- ( ph -> F : Z --> X )
txlm.g	|- ( ph -> G : Z --> Y )
lmcn2.fl	|- ( ph -> F ( ~~> t ` J ) R )
lmcn2.gl	|- ( ph -> G ( ~~> t ` K ) S )
lmcn2.o	|- ( ph -> O e. ( ( J tX K ) Cn N ) )
lmcn2.h	|- H = ( n e. Z |-> ( ( F ` n ) O ( G ` n ) ) )

Assertion

Ref	Expression
lmcn2	|- ( ph -> H ( ~~> t ` N ) ( R O S ) )

Distinct variable groups:   n, F   n, O   ph, n   n, G   n, J   n, K   n, X   n, Y   n, Z

Allowed substitution hints:   R( n)   S( n)   H( n)   M( n)   N( n)

Proof of Theorem lmcn2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		txlm.f	. . . . . . 7 |- ( ph -> F : Z --> X )
2	1	ffvelcdmda 5790	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. X )
3		txlm.g	. . . . . . 7 |- ( ph -> G : Z --> Y )
4	3	ffvelcdmda 5790	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( G ` n ) e. Y )
5	2, 4	opelxpd 4764	. . . . 5 |- ( ( ph /\ n e. Z ) -> <. ( F ` n ) , ( G ` n ) >. e. ( X X. Y ) )
6		eqidd 2232	. . . . 5 |- ( ph -> ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) )
7		txlm.j	. . . . . . . 8 |- ( ph -> J e. (TopOn ` X ) )
8		txlm.k	. . . . . . . 8 |- ( ph -> K e. (TopOn ` Y ) )
9		txtopon 15056	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J tX K ) e. (TopOn ` ( X X. Y ) ) )
10	7, 8, 9	syl2anc 411	. . . . . . 7 |- ( ph -> ( J tX K ) e. (TopOn ` ( X X. Y ) ) )
11		lmcn2.o	. . . . . . . . 9 |- ( ph -> O e. ( ( J tX K ) Cn N ) )
12		cntop2 14996	. . . . . . . . 9 |- ( O e. ( ( J tX K ) Cn N ) -> N e. Top )
13	11, 12	syl 14	. . . . . . . 8 |- ( ph -> N e. Top )
14		toptopon2 14813	. . . . . . . 8 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
15	13, 14	sylib 122	. . . . . . 7 |- ( ph -> N e. (TopOn ` U. N ) )
16		cnf2 14999	. . . . . . 7 |- ( ( ( J tX K ) e. (TopOn ` ( X X. Y ) ) /\ N e. (TopOn ` U. N ) /\ O e. ( ( J tX K ) Cn N ) ) -> O : ( X X. Y ) --> U. N )
17	10, 15, 11, 16	syl3anc 1274	. . . . . 6 |- ( ph -> O : ( X X. Y ) --> U. N )
18	17	feqmptd 5708	. . . . 5 |- ( ph -> O = ( x e. ( X X. Y ) |-> ( O ` x ) ) )
19		fveq2 5648	. . . . . 6 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( O ` <. ( F ` n ) , ( G ` n ) >. ) )
20		df-ov 6031	. . . . . 6 |- ( ( F ` n ) O ( G ` n ) ) = ( O ` <. ( F ` n ) , ( G ` n ) >. )
21	19, 20	eqtr4di 2282	. . . . 5 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( ( F ` n ) O ( G ` n ) ) )
22	5, 6, 18, 21	fmptco 5821	. . . 4 |- ( ph -> ( O o. ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ) = ( n e. Z |-> ( ( F ` n ) O ( G ` n ) ) ) )
23		lmcn2.h	. . . 4 |- H = ( n e. Z |-> ( ( F ` n ) O ( G ` n ) ) )
24	22, 23	eqtr4di 2282	. . 3 |- ( ph -> ( O o. ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ) = H )
25		lmcn2.fl	. . . . 5 |- ( ph -> F ( ~~> t ` J ) R )
26		lmcn2.gl	. . . . 5 |- ( ph -> G ( ~~> t ` K ) S )
27		txlm.z	. . . . . 6 |- Z = ( ZZ>= ` M )
28		txlm.m	. . . . . 6 |- ( ph -> M e. ZZ )
29		eqid 2231	. . . . . 6 |- ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )
30	27, 28, 7, 8, 1, 3, 29	txlm 15073	. . . . 5 |- ( ph -> ( ( F ( ~~> t ` J ) R /\ G ( ~~> t ` K ) S ) <-> ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ( ~~> t ` ( J tX K ) ) <. R , S >. ) )
31	25, 26, 30	mpbi2and 952	. . . 4 |- ( ph -> ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ( ~~> t ` ( J tX K ) ) <. R , S >. )
32	31, 11	lmcn 15045	. . 3 |- ( ph -> ( O o. ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ) ( ~~> t ` N ) ( O ` <. R , S >. ) )
33	24, 32	eqbrtrrd 4117	. 2 |- ( ph -> H ( ~~> t ` N ) ( O ` <. R , S >. ) )
34		df-ov 6031	. 2 |- ( R O S ) = ( O ` <. R , S >. )
35	33, 34	breqtrrdi 4135	1 |- ( ph -> H ( ~~> t ` N ) ( R O S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   <.cop 3676   U.cuni 3898   class class class wbr 4093   |-> cmpt 4155   X. cxp 4729   o. ccom 4735   -->wf 5329   ` cfv 5333  (class class class)co 6028   ZZcz 9523   ZZ>=cuz 9799   Topctop 14791  TopOnctopon 14804   Cn ccn 14979   ~~> tclm 14981   tX ctx 15046

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-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  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-id 4396  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-map 6862  df-pm 6863  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-topgen 13406  df-top 14792  df-topon 14805  df-bases 14837  df-cn 14982  df-cnp 14983  df-lm 14984  df-tx 15047

This theorem is referenced by: (None)