Theorem lmcn2 14962

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 5772	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. X )
3		txlm.g	. . . . . . 7 |- ( ph -> G : Z --> Y )
4	3	ffvelcdmda 5772	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( G ` n ) e. Y )
5	2, 4	opelxpd 4752	. . . . 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 14944	. . . . . . . 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 14884	. . . . . . . . 9 |- ( O e. ( ( J tX K ) Cn N ) -> N e. Top )
13	11, 12	syl 14	. . . . . . . 8 |- ( ph -> N e. Top )
14		toptopon2 14701	. . . . . . . 8 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
15	13, 14	sylib 122	. . . . . . 7 |- ( ph -> N e. (TopOn ` U. N ) )
16		cnf2 14887	. . . . . . 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 5689	. . . . 5 |- ( ph -> O = ( x e. ( X X. Y ) |-> ( O ` x ) ) )
19		fveq2 5629	. . . . . 6 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( O ` <. ( F ` n ) , ( G ` n ) >. ) )
20		df-ov 6010	. . . . . 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 5803	. . . 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 14961	. . . . 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 14933	. . 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 6010	. 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 4717   o. ccom 4723   -->wf 5314   ` cfv 5318  (class class class)co 6007   ZZcz 9454   ZZ>=cuz 9730   Topctop 14679  TopOnctopon 14692   Cn ccn 14867   ~~> tclm 14869   tX ctx 14934

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123

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 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-pm 6806  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-n0 9378  df-z 9455  df-uz 9731  df-topgen 13301  df-top 14680  df-topon 14693  df-bases 14725  df-cn 14870  df-cnp 14871  df-lm 14872  df-tx 14935

This theorem is referenced by: (None)