Theorem lmcn2 13447

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 5647	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. X )
3		txlm.g	. . . . . . 7 |- ( ph -> G : Z --> Y )
4	3	ffvelcdmda 5647	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( G ` n ) e. Y )
5	2, 4	opelxpd 4656	. . . . 5 |- ( ( ph /\ n e. Z ) -> <. ( F ` n ) , ( G ` n ) >. e. ( X X. Y ) )
6		eqidd 2178	. . . . 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 13429	. . . . . . . 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 13369	. . . . . . . . 9 |- ( O e. ( ( J tX K ) Cn N ) -> N e. Top )
13	11, 12	syl 14	. . . . . . . 8 |- ( ph -> N e. Top )
14		toptopon2 13184	. . . . . . . 8 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
15	13, 14	sylib 122	. . . . . . 7 |- ( ph -> N e. (TopOn ` U. N ) )
16		cnf2 13372	. . . . . . 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 1238	. . . . . 6 |- ( ph -> O : ( X X. Y ) --> U. N )
18	17	feqmptd 5565	. . . . 5 |- ( ph -> O = ( x e. ( X X. Y ) |-> ( O ` x ) ) )
19		fveq2 5511	. . . . . 6 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( O ` <. ( F ` n ) , ( G ` n ) >. ) )
20		df-ov 5872	. . . . . 6 |- ( ( F ` n ) O ( G ` n ) ) = ( O ` <. ( F ` n ) , ( G ` n ) >. )
21	19, 20	eqtr4di 2228	. . . . 5 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( ( F ` n ) O ( G ` n ) ) )
22	5, 6, 18, 21	fmptco 5678	. . . 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 2228	. . 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 2177	. . . . . 6 |- ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )
30	27, 28, 7, 8, 1, 3, 29	txlm 13446	. . . . 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 943	. . . 4 |- ( ph -> ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ( ~~> t ` ( J tX K ) ) <. R , S >. )
32	31, 11	lmcn 13418	. . 3 |- ( ph -> ( O o. ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ) ( ~~> t ` N ) ( O ` <. R , S >. ) )
33	24, 32	eqbrtrrd 4024	. 2 |- ( ph -> H ( ~~> t ` N ) ( O ` <. R , S >. ) )
34		df-ov 5872	. 2 |- ( R O S ) = ( O ` <. R , S >. )
35	33, 34	breqtrrdi 4042	1 |- ( ph -> H ( ~~> t ` N ) ( R O S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   <.cop 3594   U.cuni 3807   class class class wbr 4000   |-> cmpt 4061   X. cxp 4621   o. ccom 4627   -->wf 5208   ` cfv 5212  (class class class)co 5869   ZZcz 9242   ZZ>=cuz 9517   Topctop 13162  TopOnctopon 13175   Cn ccn 13352   ~~> tclm 13354   tX ctx 13419

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pm 6645  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-topgen 12657  df-top 13163  df-topon 13176  df-bases 13208  df-cn 13355  df-cnp 13356  df-lm 13357  df-tx 13420

This theorem is referenced by: (None)