Theorem lmcn2 14975

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 5775	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. X )
3		txlm.g	. . . . . . 7 |- ( ph -> G : Z --> Y )
4	3	ffvelcdmda 5775	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( G ` n ) e. Y )
5	2, 4	opelxpd 4753	. . . . 5 |- ( ( ph /\ n e. Z ) -> <. ( F ` n ) , ( G ` n ) >. e. ( X X. Y ) )
6		eqidd 2230	. . . . 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 14957	. . . . . . . 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 14897	. . . . . . . . 9 |- ( O e. ( ( J tX K ) Cn N ) -> N e. Top )
13	11, 12	syl 14	. . . . . . . 8 |- ( ph -> N e. Top )
14		toptopon2 14714	. . . . . . . 8 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
15	13, 14	sylib 122	. . . . . . 7 |- ( ph -> N e. (TopOn ` U. N ) )
16		cnf2 14900	. . . . . . 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 1271	. . . . . 6 |- ( ph -> O : ( X X. Y ) --> U. N )
18	17	feqmptd 5692	. . . . 5 |- ( ph -> O = ( x e. ( X X. Y ) |-> ( O ` x ) ) )
19		fveq2 5632	. . . . . 6 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( O ` <. ( F ` n ) , ( G ` n ) >. ) )
20		df-ov 6013	. . . . . 6 |- ( ( F ` n ) O ( G ` n ) ) = ( O ` <. ( F ` n ) , ( G ` n ) >. )
21	19, 20	eqtr4di 2280	. . . . 5 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( ( F ` n ) O ( G ` n ) ) )
22	5, 6, 18, 21	fmptco 5806	. . . 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 2280	. . 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 2229	. . . . . 6 |- ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )
30	27, 28, 7, 8, 1, 3, 29	txlm 14974	. . . . 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 949	. . . 4 |- ( ph -> ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ( ~~> t ` ( J tX K ) ) <. R , S >. )
32	31, 11	lmcn 14946	. . 3 |- ( ph -> ( O o. ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ) ( ~~> t ` N ) ( O ` <. R , S >. ) )
33	24, 32	eqbrtrrd 4107	. 2 |- ( ph -> H ( ~~> t ` N ) ( O ` <. R , S >. ) )
34		df-ov 6013	. 2 |- ( R O S ) = ( O ` <. R , S >. )
35	33, 34	breqtrrdi 4125	1 |- ( ph -> H ( ~~> t ` N ) ( R O S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   <.cop 3669   U.cuni 3888   class class class wbr 4083   |-> cmpt 4145   X. cxp 4718   o. ccom 4724   -->wf 5317   ` cfv 5321  (class class class)co 6010   ZZcz 9462   ZZ>=cuz 9738   Topctop 14692  TopOnctopon 14705   Cn ccn 14880   ~~> tclm 14882   tX ctx 14947

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-map 6810  df-pm 6811  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-topgen 13314  df-top 14693  df-topon 14706  df-bases 14738  df-cn 14883  df-cnp 14884  df-lm 14885  df-tx 14948

This theorem is referenced by: (None)