Theorem xkofvcn 21487

Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21459.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
xkofvcn.1	|- X = U. R
xkofvcn.2	|- F = ( f e. ( R Cn S ) , x e. X |-> ( f ` x ) )

Assertion

Ref	Expression
xkofvcn	|- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> F e. ( ( ( S ^ko R ) tX R ) Cn S ) )

Distinct variable groups:   x, f, R   S, f, x   f, X, x

Allowed substitution hints:   F( x, f)

Proof of Theorem xkofvcn

Dummy variables g h y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xkofvcn.2	. 2 |- F = ( f e. ( R Cn S ) , x e. X |-> ( f ` x ) )
2		nllytop 21276	. . . 4 |- ( R e. 𝑛Locally Comp -> R e. Top )
3		eqid 2622	. . . . 5 |- ( S ^ko R ) = ( S ^ko R )
4	3	xkotopon 21403	. . . 4 |- ( ( R e. Top /\ S e. Top ) -> ( S ^ko R ) e. (TopOn ` ( R Cn S ) ) )
5	2, 4	sylan 488	. . 3 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( S ^ko R ) e. (TopOn ` ( R Cn S ) ) )
6	2	adantr 481	. . . 4 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> R e. Top )
7		xkofvcn.1	. . . . 5 |- X = U. R
8	7	toptopon 20722	. . . 4 |- ( R e. Top <-> R e. (TopOn ` X ) )
9	6, 8	sylib 208	. . 3 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> R e. (TopOn ` X ) )
10	5, 9	cnmpt1st 21471	. . . 4 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( f e. ( R Cn S ) , x e. X |-> f ) e. ( ( ( S ^ko R ) tX R ) Cn ( S ^ko R ) ) )
11	5, 9	cnmpt2nd 21472	. . . . 5 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( f e. ( R Cn S ) , x e. X |-> x ) e. ( ( ( S ^ko R ) tX R ) Cn R ) )
12		1on 7567	. . . . . . 7 |- 1o e. On
13		distopon 20801	. . . . . . 7 |- ( 1o e. On -> ~P 1o e. (TopOn ` 1o ) )
14	12, 13	mp1i 13	. . . . . 6 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ~P 1o e. (TopOn ` 1o ) )
15		xkoccn 21422	. . . . . 6 |- ( ( ~P 1o e. (TopOn ` 1o ) /\ R e. (TopOn ` X ) ) -> ( y e. X |-> ( 1o X. { y } ) ) e. ( R Cn ( R ^ko ~P 1o ) ) )
16	14, 9, 15	syl2anc 693	. . . . 5 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( y e. X |-> ( 1o X. { y } ) ) e. ( R Cn ( R ^ko ~P 1o ) ) )
17		sneq 4187	. . . . . 6 |- ( y = x -> { y } = { x } )
18	17	xpeq2d 5139	. . . . 5 |- ( y = x -> ( 1o X. { y } ) = ( 1o X. { x } ) )
19	5, 9, 11, 9, 16, 18	cnmpt21 21474	. . . 4 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( f e. ( R Cn S ) , x e. X |-> ( 1o X. { x } ) ) e. ( ( ( S ^ko R ) tX R ) Cn ( R ^ko ~P 1o ) ) )
20		distop 20799	. . . . . 6 |- ( 1o e. On -> ~P 1o e. Top )
21	12, 20	mp1i 13	. . . . 5 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ~P 1o e. Top )
22		eqid 2622	. . . . . 6 |- ( R ^ko ~P 1o ) = ( R ^ko ~P 1o )
23	22	xkotopon 21403	. . . . 5 |- ( ( ~P 1o e. Top /\ R e. Top ) -> ( R ^ko ~P 1o ) e. (TopOn ` ( ~P 1o Cn R ) ) )
24	21, 6, 23	syl2anc 693	. . . 4 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( R ^ko ~P 1o ) e. (TopOn ` ( ~P 1o Cn R ) ) )
25		simpl 473	. . . . 5 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> R e. 𝑛Locally Comp )
26		simpr 477	. . . . 5 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> S e. Top )
27		eqid 2622	. . . . . 6 |- ( g e. ( R Cn S ) , h e. ( ~P 1o Cn R ) |-> ( g o. h ) ) = ( g e. ( R Cn S ) , h e. ( ~P 1o Cn R ) |-> ( g o. h ) )
28	27	xkococn 21463	. . . . 5 |- ( ( ~P 1o e. Top /\ R e. 𝑛Locally Comp /\ S e. Top ) -> ( g e. ( R Cn S ) , h e. ( ~P 1o Cn R ) |-> ( g o. h ) ) e. ( ( ( S ^ko R ) tX ( R ^ko ~P 1o ) ) Cn ( S ^ko ~P 1o ) ) )
29	21, 25, 26, 28	syl3anc 1326	. . . 4 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( g e. ( R Cn S ) , h e. ( ~P 1o Cn R ) |-> ( g o. h ) ) e. ( ( ( S ^ko R ) tX ( R ^ko ~P 1o ) ) Cn ( S ^ko ~P 1o ) ) )
30		coeq1 5279	. . . . 5 |- ( g = f -> ( g o. h ) = ( f o. h ) )
31		coeq2 5280	. . . . 5 |- ( h = ( 1o X. { x } ) -> ( f o. h ) = ( f o. ( 1o X. { x } ) ) )
32	30, 31	sylan9eq 2676	. . . 4 |- ( ( g = f /\ h = ( 1o X. { x } ) ) -> ( g o. h ) = ( f o. ( 1o X. { x } ) ) )
33	5, 9, 10, 19, 5, 24, 29, 32	cnmpt22 21477	. . 3 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( f e. ( R Cn S ) , x e. X |-> ( f o. ( 1o X. { x } ) ) ) e. ( ( ( S ^ko R ) tX R ) Cn ( S ^ko ~P 1o ) ) )
34		eqid 2622	. . . . 5 |- ( S ^ko ~P 1o ) = ( S ^ko ~P 1o )
35	34	xkotopon 21403	. . . 4 |- ( ( ~P 1o e. Top /\ S e. Top ) -> ( S ^ko ~P 1o ) e. (TopOn ` ( ~P 1o Cn S ) ) )
36	21, 26, 35	syl2anc 693	. . 3 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( S ^ko ~P 1o ) e. (TopOn ` ( ~P 1o Cn S ) ) )
37		0lt1o 7584	. . . . 5 |- (/) e. 1o
38	37	a1i 11	. . . 4 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> (/) e. 1o )
39		unipw 4918	. . . . . 6 |- U. ~P 1o = 1o
40	39	eqcomi 2631	. . . . 5 |- 1o = U. ~P 1o
41	40	xkopjcn 21459	. . . 4 |- ( ( ~P 1o e. Top /\ S e. Top /\ (/) e. 1o ) -> ( g e. ( ~P 1o Cn S ) |-> ( g ` (/) ) ) e. ( ( S ^ko ~P 1o ) Cn S ) )
42	21, 26, 38, 41	syl3anc 1326	. . 3 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( g e. ( ~P 1o Cn S ) |-> ( g ` (/) ) ) e. ( ( S ^ko ~P 1o ) Cn S ) )
43		fveq1 6190	. . . 4 |- ( g = ( f o. ( 1o X. { x } ) ) -> ( g ` (/) ) = ( ( f o. ( 1o X. { x } ) ) ` (/) ) )
44		vex 3203	. . . . . . 7 |- x e. _V
45	44	fconst 6091	. . . . . 6 |- ( 1o X. { x } ) : 1o --> { x }
46		fvco3 6275	. . . . . 6 |- ( ( ( 1o X. { x } ) : 1o --> { x } /\ (/) e. 1o ) -> ( ( f o. ( 1o X. { x } ) ) ` (/) ) = ( f ` ( ( 1o X. { x } ) ` (/) ) ) )
47	45, 37, 46	mp2an 708	. . . . 5 |- ( ( f o. ( 1o X. { x } ) ) ` (/) ) = ( f ` ( ( 1o X. { x } ) ` (/) ) )
48	44	fvconst2 6469	. . . . . . 7 |- ( (/) e. 1o -> ( ( 1o X. { x } ) ` (/) ) = x )
49	37, 48	ax-mp 5	. . . . . 6 |- ( ( 1o X. { x } ) ` (/) ) = x
50	49	fveq2i 6194	. . . . 5 |- ( f ` ( ( 1o X. { x } ) ` (/) ) ) = ( f ` x )
51	47, 50	eqtri 2644	. . . 4 |- ( ( f o. ( 1o X. { x } ) ) ` (/) ) = ( f ` x )
52	43, 51	syl6eq 2672	. . 3 |- ( g = ( f o. ( 1o X. { x } ) ) -> ( g ` (/) ) = ( f ` x ) )
53	5, 9, 33, 36, 42, 52	cnmpt21 21474	. 2 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> ( f e. ( R Cn S ) , x e. X |-> ( f ` x ) ) e. ( ( ( S ^ko R ) tX R ) Cn S ) )
54	1, 53	syl5eqel 2705	1 |- ( ( R e. 𝑛Locally Comp /\ S e. Top ) -> F e. ( ( ( S ^ko R ) tX R ) Cn S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   Compccmp 21189  𝑛Locally cnlly 21268   tX ctx 21363   ^ko cxko 21364

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-3or 1038  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-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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-cnp 21032  df-cmp 21190  df-nlly 21270  df-tx 21365  df-xko 21366

This theorem is referenced by:  cnmptk1p  21488  cnmptk2  21489