Theorem oprcn 7977

Description: Construct a continuous function H built from two functions F and G (on a common metric space A) applied to an operation O. This can be used to show, e.g., that x^2 + x is continuous if we know that x^2, x, and + are.

Hypotheses

Ref	Expression
oprcn.1	|- X = dom dom A
oprcn.2	|- Y = dom dom B
oprcn.4	|- Z = dom dom C
oprcn.6	|- A e. Met
oprcn.7	|- B e. Met
oprcn.8	|- C e. Met
oprcn.9	|- J e. Met
oprcn.a	|- K = (Open` A)
oprcn.b	|- L = (Open` B)
oprcn.c	|- M = (Open` C)
oprcn.d	|- N = (Open` D)
oprcn.j	|- Q = (Open` J)
oprcn.10	|- D = {<.<.x, y>., z>. | ((x e. (Y X. Z) /\ y e. (Y X. Z)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
oprcn.11	|- O e. (N Cn Q)
oprcn.12	|- H = {<.w, v>. | (w e. X /\ v = ((F` w)O(G` w)))}

Assertion

Ref	Expression
oprcn	|- ((F e. (K Cn L) /\ G e. (K Cn M)) -> H e. (K Cn Q))

Distinct variable groups:   w,A   x,w,y,z,B   w,C,x,y,z   w,v,x,y,z,F   v,G,w,x,y,z   w,J   w,K   w,L   w,M   v,O,w   v,X,w,x,y,z   v,Y,w,x,y,z   v,Z,w,x,y,z

Proof of Theorem oprcn

Step	Hyp	Ref	Expression
1		ffvelrn 3814	. . . . . . . . . 10 |- ((F:X-->Y /\ w e. X) -> (F` w) e. Y)
2		oprcn.6	. . . . . . . . . . 11 |- A e. Met
3		oprcn.7	. . . . . . . . . . 11 |- B e. Met
4		oprcn.1	. . . . . . . . . . . 12 |- X = dom dom A
5		oprcn.2	. . . . . . . . . . . 12 |- Y = dom dom B
6		oprcn.a	. . . . . . . . . . . 12 |- K = (Open` A)
7		oprcn.b	. . . . . . . . . . . 12 |- L = (Open` B)
8	4, 5, 6, 7	metcnf 7884	. . . . . . . . . . 11 |- ((A e. Met /\ B e. Met /\ F e. (K Cn L)) -> F:X-->Y)
9	2, 3, 8	mp3an12 906	. . . . . . . . . 10 |- (F e. (K Cn L) -> F:X-->Y)
10	1, 9	sylan 448	. . . . . . . . 9 |- ((F e. (K Cn L) /\ w e. X) -> (F` w) e. Y)
11		ffvelrn 3814	. . . . . . . . . 10 |- ((G:X-->Z /\ w e. X) -> (G` w) e. Z)
12		oprcn.8	. . . . . . . . . . 11 |- C e. Met
13		oprcn.4	. . . . . . . . . . . 12 |- Z = dom dom C
14		oprcn.c	. . . . . . . . . . . 12 |- M = (Open` C)
15	4, 13, 6, 14	metcnf 7884	. . . . . . . . . . 11 |- ((A e. Met /\ C e. Met /\ G e. (K Cn M)) -> G:X-->Z)
16	2, 12, 15	mp3an12 906	. . . . . . . . . 10 |- (G e. (K Cn M) -> G:X-->Z)
17	11, 16	sylan 448	. . . . . . . . 9 |- ((G e. (K Cn M) /\ w e. X) -> (G` w) e. Z)
18	10, 17	anim12i 333	. . . . . . . 8 |- (((F e. (K Cn L) /\ w e. X) /\ (G e. (K Cn M) /\ w e. X)) -> ((F` w) e. Y /\ (G` w) e. Z))
19	18	anandirs 513	. . . . . . 7 |- (((F e. (K Cn L) /\ G e. (K Cn M)) /\ w e. X) -> ((F` w) e. Y /\ (G` w) e. Z))
20		opelxpi 3217	. . . . . . 7 |- (((F` w) e. Y /\ (G` w) e. Z) -> <.(F` w), (G` w)>. e. (Y X. Z))
21	19, 20	syl 10	. . . . . 6 |- (((F e. (K Cn L) /\ G e. (K Cn M)) /\ w e. X) -> <.(F` w), (G` w)>. e. (Y X. Z))
22	21	r19.21aiva 1714	. . . . 5 |- ((F e. (K Cn L) /\ G e. (K Cn M)) -> A.w e. X <.(F` w), (G` w)>. e. (Y X. Z))
23		eqid 1475	. . . . . 6 |- {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)} = {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)}
24		opex 2782	. . . . . 6 |- <.(F` w), (G` w)>. e. V
25	23, 24	rnssopab 3825	. . . . 5 |- (A.w e. X <.(F` w), (G` w)>. e. (Y X. Z) <-> ran {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)} (_ (Y X. Z))
26	22, 25	sylib 198	. . . 4 |- ((F e. (K Cn L) /\ G e. (K Cn M)) -> ran {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)} (_ (Y X. Z))
27		fvex 3732	. . . . 5 |- (O` w) e. V
28		oprcn.10	. . . . . . . 8 |- D = {<.<.x, y>., z>. | ((x e. (Y X. Z) /\ y e. (Y X. Z)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
29	5, 13, 3, 12, 28	metxp 7834	. . . . . . 7 |- D e. Met
30		oprcn.9	. . . . . . 7 |- J e. Met
31		oprcn.11	. . . . . . 7 |- O e. (N Cn Q)
32		ltso 5512	. . . . . . . . . . . 12 |- < Or RR
33	32	supex 4577	. . . . . . . . . . 11 |- sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) e. V
34	33, 28	dmoprab2 4123	. . . . . . . . . 10 |- dom D = ((Y X. Z) X. (Y X. Z))
35	34	dmeqi 3312	. . . . . . . . 9 |- dom dom D = dom ((Y X. Z) X. (Y X. Z))
36		dmxpid 3333	. . . . . . . . 9 |- dom ((Y X. Z) X. (Y X. Z)) = (Y X. Z)
37	35, 36	eqtr2 1496	. . . . . . . 8 |- (Y X. Z) = dom dom D
38		eqid 1475	. . . . . . . 8 |- dom dom J = dom dom J
39		oprcn.d	. . . . . . . 8 |- N = (Open` D)
40		oprcn.j	. . . . . . . 8 |- Q = (Open` J)
41	37, 38, 39, 40	metcnf 7884	. . . . . . 7 |- ((D e. Met /\ J e. Met /\ O e. (N Cn Q)) -> O:(Y X. Z)-->dom dom J)
42	29, 30, 31, 41	mp3an 916	. . . . . 6 |- O:(Y X. Z)-->dom dom J
43		fopabfv 3831	. . . . . . 7 |- (O:(Y X. Z)-->dom dom J <-> (O = {<.w, v>. | (w e. (Y X. Z) /\ v = (O` w))} /\ A.w e. (Y X. Z)(O` w) e. dom dom J))
44	43	pm3.26bi 322	. . . . . 6 |- (O:(Y X. Z)-->dom dom J -> O = {<.w, v>. | (w e. (Y X. Z) /\ v = (O` w))})
45	42, 44	ax-mp 7	. . . . 5 |- O = {<.w, v>. | (w e. (Y X. Z) /\ v = (O` w))}
46	27, 24, 45, 23	fopabcos 3833	. . . 4 |- (ran {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)} (_ (Y X. Z) -> (O o. {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)}) = {<.w, v>. | (w e. X /\ v = [_<.(F` w), (G` w)>. / w]_(O` w))})
47	26, 46	syl 10	. . 3 |- ((F e. (K Cn L) /\ G e. (K Cn M)) -> (O o. {<.w, v>. | (w e. X /\ v = <.(F` w), (G` w)>.)}) = {<.w, v>. | (w e. X /\ v = [_<.(F` w), (G` w)>. / w]_(O` w))})
48		oprcn.12	. . . 4 |- H = {<.w, v>. | (w e. X /\ v = ((F` w)O(G` w)))}
49		csbvarg 2021	. . . . . . . . . 10 |- (<.(F` w), (G` w)>. e. V -> [_<.(F` w), (G` w)>. / w]_w = <.(F` w), (G` w)>.)
50	24, 49	ax-mp 7	. . . . . . . . 9 |- [_<.(F` w), (G` w)>. / w]_w = <.(F` w), (G` w)>.
51	50	fveq2i 3727	. . . . . . . 8 |- (O` [_<.(F` w), (G` w)>. / w]_w) = (O` <.(F` w), (G` w)>.)
52		csbfv2g 3743	. . . . . . . . 9 |- (<.(F` w), (G` w)>. e. V -> [_<.(F` w), (G` w)>. / w]_(O` w) = (O` [_<.(F` w), (G` w)>. / w]_w))
53	24, 52	ax-mp 7	. . . . . . . 8 |- [_<.(F` w), (G` w)>. / w]_(O` w) = (O` [_<.(F` w), (G` w)>. / w]_w)
54		df-opr 3965	. . . . . . . 8 |- ((F` w)O(G` w)) = (O` <.(F` w), (G` w)>.)
55	51, 53, 54	3eqtr4r 1506	. . . . . . 7 |- ((F` w)O(G` w)) = [_<.(F` w), (G` w)>. / w]_(O` w)
56	55	eqeq2i 1485	. . . . . 6 |- (v = ((F` w)O(G` w)) <-> v = [_<.(F` w), (G` w)>. / w]_(O` w))
57	56	anbi2i 480	. . . . 5 |- ((w e. X /\ v = ((F` w)O(G` w))) <-> (w e. X /\ v = [_<.(F` w), (G` w)>. / w]_(O` w)))
58	57	opabbii 2671	. . . 4 |- {<.w, v>. | (w e. X /\ v = ((F` w)O(G` w)))} = {<.w, v>. | (w e. X /\ v = [_<.(F` w), (G` w)>. / w]_(O` w))}
59	48, 58	eqtr 1495	. . 3 |- H = {<.w, v>. | <