Theorem flfcnp2 21811

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, 19-Sep-2015.)

Hypotheses

Ref	Expression
flfcnp2.j	|- ( ph -> J e. (TopOn ` X ) )
flfcnp2.k	|- ( ph -> K e. (TopOn ` Y ) )
flfcnp2.l	|- ( ph -> L e. ( Fil ` Z ) )
flfcnp2.a	|- ( ( ph /\ x e. Z ) -> A e. X )
flfcnp2.b	|- ( ( ph /\ x e. Z ) -> B e. Y )
flfcnp2.r	|- ( ph -> R e. ( ( J fLimf L ) ` ( x e. Z |-> A ) ) )
flfcnp2.s	|- ( ph -> S e. ( ( K fLimf L ) ` ( x e. Z |-> B ) ) )
flfcnp2.o	|- ( ph -> O e. ( ( ( J tX K ) CnP N ) ` <. R , S >. ) )

Assertion

Ref	Expression
flfcnp2	|- ( ph -> ( R O S ) e. ( ( N fLimf L ) ` ( x e. Z |-> ( A O B ) ) ) )

Distinct variable groups:   x, O   ph, x   x, Z   x, X   x, Y

Allowed substitution hints:   A( x)   B( x)   R( x)   S( x)   J( x)   K( x)   L( x)   N( x)

Proof of Theorem flfcnp2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( R O S ) = ( O ` <. R , S >. )
2		flfcnp2.j	. . . . 5 |- ( ph -> J e. (TopOn ` X ) )
3		flfcnp2.k	. . . . 5 |- ( ph -> K e. (TopOn ` Y ) )
4		txtopon 21394	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J tX K ) e. (TopOn ` ( X X. Y ) ) )
5	2, 3, 4	syl2anc 693	. . . 4 |- ( ph -> ( J tX K ) e. (TopOn ` ( X X. Y ) ) )
6		flfcnp2.l	. . . 4 |- ( ph -> L e. ( Fil ` Z ) )
7		flfcnp2.a	. . . . . 6 |- ( ( ph /\ x e. Z ) -> A e. X )
8		flfcnp2.b	. . . . . 6 |- ( ( ph /\ x e. Z ) -> B e. Y )
9		opelxpi 5148	. . . . . 6 |- ( ( A e. X /\ B e. Y ) -> <. A , B >. e. ( X X. Y ) )
10	7, 8, 9	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. Z ) -> <. A , B >. e. ( X X. Y ) )
11		eqid 2622	. . . . 5 |- ( x e. Z |-> <. A , B >. ) = ( x e. Z |-> <. A , B >. )
12	10, 11	fmptd 6385	. . . 4 |- ( ph -> ( x e. Z |-> <. A , B >. ) : Z --> ( X X. Y ) )
13		flfcnp2.r	. . . . . 6 |- ( ph -> R e. ( ( J fLimf L ) ` ( x e. Z |-> A ) ) )
14		flfcnp2.s	. . . . . 6 |- ( ph -> S e. ( ( K fLimf L ) ` ( x e. Z |-> B ) ) )
15		eqid 2622	. . . . . . . 8 |- ( x e. Z |-> A ) = ( x e. Z |-> A )
16	7, 15	fmptd 6385	. . . . . . 7 |- ( ph -> ( x e. Z |-> A ) : Z --> X )
17		eqid 2622	. . . . . . . 8 |- ( x e. Z |-> B ) = ( x e. Z |-> B )
18	8, 17	fmptd 6385	. . . . . . 7 |- ( ph -> ( x e. Z |-> B ) : Z --> Y )
19		nfcv 2764	. . . . . . . 8 |- F/_ y <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >.
20		nffvmpt1 6199	. . . . . . . . 9 |- F/_ x ( ( x e. Z |-> A ) ` y )
21		nffvmpt1 6199	. . . . . . . . 9 |- F/_ x ( ( x e. Z |-> B ) ` y )
22	20, 21	nfop 4418	. . . . . . . 8 |- F/_ x <. ( ( x e. Z |-> A ) ` y ) , ( ( x e. Z |-> B ) ` y ) >.
23		fveq2 6191	. . . . . . . . 9 |- ( x = y -> ( ( x e. Z |-> A ) ` x ) = ( ( x e. Z |-> A ) ` y ) )
24		fveq2 6191	. . . . . . . . 9 |- ( x = y -> ( ( x e. Z |-> B ) ` x ) = ( ( x e. Z |-> B ) ` y ) )
25	23, 24	opeq12d 4410	. . . . . . . 8 |- ( x = y -> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. = <. ( ( x e. Z |-> A ) ` y ) , ( ( x e. Z |-> B ) ` y ) >. )
26	19, 22, 25	cbvmpt 4749	. . . . . . 7 |- ( x e. Z |-> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. ) = ( y e. Z |-> <. ( ( x e. Z |-> A ) ` y ) , ( ( x e. Z |-> B ) ` y ) >. )
27	2, 3, 6, 16, 18, 26	txflf 21810	. . . . . 6 |- ( ph -> ( <. R , S >. e. ( ( ( J tX K ) fLimf L ) ` ( x e. Z |-> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. ) ) <-> ( R e. ( ( J fLimf L ) ` ( x e. Z |-> A ) ) /\ S e. ( ( K fLimf L ) ` ( x e. Z |-> B ) ) ) ) )
28	13, 14, 27	mpbir2and 957	. . . . 5 |- ( ph -> <. R , S >. e. ( ( ( J tX K ) fLimf L ) ` ( x e. Z |-> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. ) ) )
29		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. Z ) -> x e. Z )
30	15	fvmpt2 6291	. . . . . . . . 9 |- ( ( x e. Z /\ A e. X ) -> ( ( x e. Z |-> A ) ` x ) = A )
31	29, 7, 30	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. Z ) -> ( ( x e. Z |-> A ) ` x ) = A )
32	17	fvmpt2 6291	. . . . . . . . 9 |- ( ( x e. Z /\ B e. Y ) -> ( ( x e. Z |-> B ) ` x ) = B )
33	29, 8, 32	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. Z ) -> ( ( x e. Z |-> B ) ` x ) = B )
34	31, 33	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ x e. Z ) -> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. = <. A , B >. )
35	34	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( x e. Z |-> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. ) = ( x e. Z |-> <. A , B >. ) )
36	35	fveq2d 6195	. . . . 5 |- ( ph -> ( ( ( J tX K ) fLimf L ) ` ( x e. Z |-> <. ( ( x e. Z |-> A ) ` x ) , ( ( x e. Z |-> B ) ` x ) >. ) ) = ( ( ( J tX K ) fLimf L ) ` ( x e. Z |-> <. A , B >. ) ) )
37	28, 36	eleqtrd 2703	. . . 4 |- ( ph -> <. R , S >. e. ( ( ( J tX K ) fLimf L ) ` ( x e. Z |-> <. A , B >. ) ) )
38		flfcnp2.o	. . . 4 |- ( ph -> O e. ( ( ( J tX K ) CnP N ) ` <. R , S >. ) )
39		flfcnp 21808	. . . 4 |- ( ( ( ( J tX K ) e. (TopOn ` ( X X. Y ) ) /\ L e. ( Fil ` Z ) /\ ( x e. Z |-> <. A , B >. ) : Z --> ( X X. Y ) ) /\ ( <. R , S >. e. ( ( ( J tX K ) fLimf L ) ` ( x e. Z |-> <. A , B >. ) ) /\ O e. ( ( ( J tX K ) CnP N ) ` <. R , S >. ) ) ) -> ( O ` <. R , S >. ) e. ( ( N fLimf L ) ` ( O o. ( x e. Z |-> <. A , B >. ) ) ) )
40	5, 6, 12, 37, 38, 39	syl32anc 1334	. . 3 |- ( ph -> ( O ` <. R , S >. ) e. ( ( N fLimf L ) ` ( O o. ( x e. Z |-> <. A , B >. ) ) ) )
41		eqidd 2623	. . . . 5 |- ( ph -> ( x e. Z |-> <. A , B >. ) = ( x e. Z |-> <. A , B >. ) )
42		cnptop2 21047	. . . . . . . . 9 |- ( O e. ( ( ( J tX K ) CnP N ) ` <. R , S >. ) -> N e. Top )
43	38, 42	syl 17	. . . . . . . 8 |- ( ph -> N e. Top )
44		eqid 2622	. . . . . . . . 9 |- U. N = U. N
45	44	toptopon 20722	. . . . . . . 8 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
46	43, 45	sylib 208	. . . . . . 7 |- ( ph -> N e. (TopOn ` U. N ) )
47		cnpf2 21054	. . . . . . 7 |- ( ( ( J tX K ) e. (TopOn ` ( X X. Y ) ) /\ N e. (TopOn ` U. N ) /\ O e. ( ( ( J tX K ) CnP N ) ` <. R , S >. ) ) -> O : ( X X. Y ) --> U. N )
48	5, 46, 38, 47	syl3anc 1326	. . . . . 6 |- ( ph -> O : ( X X. Y ) --> U. N )
49	48	feqmptd 6249	. . . . 5 |- ( ph -> O = ( y e. ( X X. Y ) |-> ( O ` y ) ) )
50		fveq2 6191	. . . . . 6 |- ( y = <. A , B >. -> ( O ` y ) = ( O ` <. A , B >. ) )
51		df-ov 6653	. . . . . 6 |- ( A O B ) = ( O ` <. A , B >. )
52	50, 51	syl6eqr 2674	. . . . 5 |- ( y = <. A , B >. -> ( O ` y ) = ( A O B ) )
53	10, 41, 49, 52	fmptco 6396	. . . 4 |- ( ph -> ( O o. ( x e. Z |-> <. A , B >. ) ) = ( x e. Z |-> ( A O B ) ) )
54	53	fveq2d 6195	. . 3 |- ( ph -> ( ( N fLimf L ) ` ( O o. ( x e. Z |-> <. A , B >. ) ) ) = ( ( N fLimf L ) ` ( x e. Z |-> ( A O B ) ) ) )
55	40, 54	eleqtrd 2703	. 2 |- ( ph -> ( O ` <. R , S >. ) e. ( ( N fLimf L ) ` ( x e. Z |-> ( A O B ) ) ) )
56	1, 55	syl5eqel 2705	1 |- ( ph -> ( R O S ) e. ( ( N fLimf L ) ` ( x e. Z |-> ( A O B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   CnP ccnp 21029   tX ctx 21363   Filcfil 21649   fLimf cflf 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-ntr 20824  df-nei 20902  df-cnp 21032  df-tx 21365  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744

This theorem is referenced by:  tsmsadd  21950