Theorem lmcn2 15506

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 5846	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. X )
3		txlm.g	. . . . . . 7 |- ( ph -> G : Z --> Y )
4	3	ffvelcdmda 5846	. . . . . 6 |- ( ( ph /\ n e. Z ) -> ( G ` n ) e. Y )
5	2, 4	opelxpd 4809	. . . . 5 |- ( ( ph /\ n e. Z ) -> <. ( F ` n ) , ( G ` n ) >. e. ( X X. Y ) )
6		eqidd 2239	. . . . 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 15488	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J tX K ) e. (TopOn ` ( X X. Y ) ) )
10	7, 8, 9	syl2anc 415	. . . . . . 7 |- ( ph -> ( J tX K ) e. (TopOn ` ( X X. Y ) ) )
11		lmcn2.o	. . . . . . . . 9 |- ( ph -> O e. ( ( J tX K ) Cn N ) )
12		cntop2 15428	. . . . . . . . 9 |- ( O e. ( ( J tX K ) Cn N ) -> N e. Top )
13	11, 12	syl 14	. . . . . . . 8 |- ( ph -> N e. Top )
14		toptopon2 15245	. . . . . . . 8 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
15	13, 14	sylib 122	. . . . . . 7 |- ( ph -> N e. (TopOn ` U. N ) )
16		cnf2 15431	. . . . . . 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 1278	. . . . . 6 |- ( ph -> O : ( X X. Y ) --> U. N )
18	17	feqmptd 5759	. . . . 5 |- ( ph -> O = ( x e. ( X X. Y ) |-> ( O ` x ) ) )
19		fveq2 5698	. . . . . 6 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( O ` <. ( F ` n ) , ( G ` n ) >. ) )
20		df-ov 6091	. . . . . 6 |- ( ( F ` n ) O ( G ` n ) ) = ( O ` <. ( F ` n ) , ( G ` n ) >. )
21	19, 20	eqtr4di 2289	. . . . 5 |- ( x = <. ( F ` n ) , ( G ` n ) >. -> ( O ` x ) = ( ( F ` n ) O ( G ` n ) ) )
22	5, 6, 18, 21	fmptco 5877	. . . 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 2289	. . 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 2238	. . . . . 6 |- ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )
30	27, 28, 7, 8, 1, 3, 29	txlm 15505	. . . . 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 956	. . . 4 |- ( ph -> ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ( ~~> t ` ( J tX K ) ) <. R , S >. )
32	31, 11	lmcn 15477	. . 3 |- ( ph -> ( O o. ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. ) ) ( ~~> t ` N ) ( O ` <. R , S >. ) )
33	24, 32	eqbrtrrd 4155	. 2 |- ( ph -> H ( ~~> t ` N ) ( O ` <. R , S >. ) )
34		df-ov 6091	. 2 |- ( R O S ) = ( O ` <. R , S >. )
35	33, 34	breqtrrdi 4173	1 |- ( ph -> H ( ~~> t ` N ) ( R O S ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3712   U.cuni 3935   class class class wbr 4131   |-> cmpt 4193   X. cxp 4774   o. ccom 4780   -->wf 5376   ` cfv 5380  (class class class)co 6088   ZZcz 9655   ZZ>=cuz 9937   Topctop 15223  TopOnctopon 15236   Cn ccn 15411   ~~> tclm 15413   tX ctx 15478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-map 6928  df-pm 6929  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-topgen 13673  df-top 15224  df-topon 15237  df-bases 15269  df-cn 15414  df-cnp 15415  df-lm 15416  df-tx 15479

This theorem is used by: (None)