Theorem kqnrmlem2 21547

Description: If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	|- F = ( x e. X |-> { y e. J | x e. y } )

Assertion

Ref	Expression
kqnrmlem2	|- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) -> J e. Nrm )

Distinct variable groups:   x, y, J   x, X, y

Allowed substitution hints:   F( x, y)

Proof of Theorem kqnrmlem2

Dummy variables m w z u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 20718	. . 3 |- ( J e. (TopOn ` X ) -> J e. Top )
2	1	adantr 481	. 2 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) -> J e. Top )
3		simplr 792	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> (KQ ` J ) e. Nrm )
4		simpll 790	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> J e. (TopOn ` X ) )
5		simprl 794	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> z e. J )
6		kqval.2	. . . . . . 7 |- F = ( x e. X |-> { y e. J | x e. y } )
7	6	kqopn 21537	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ z e. J ) -> ( F " z ) e. (KQ ` J ) )
8	4, 5, 7	syl2anc 693	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> ( F " z ) e. (KQ ` J ) )
9		inss1 3833	. . . . . . 7 |- ( ( Clsd ` J ) i^i ~P z ) C_ ( Clsd ` J )
10		simprr 796	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> w e. ( ( Clsd ` J ) i^i ~P z ) )
11	9, 10	sseldi 3601	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> w e. ( Clsd ` J ) )
12	6	kqcld 21538	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ w e. ( Clsd ` J ) ) -> ( F " w ) e. ( Clsd ` (KQ ` J ) ) )
13	4, 11, 12	syl2anc 693	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> ( F " w ) e. ( Clsd ` (KQ ` J ) ) )
14		inss2 3834	. . . . . . 7 |- ( ( Clsd ` J ) i^i ~P z ) C_ ~P z
15	14, 10	sseldi 3601	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> w e. ~P z )
16		elpwi 4168	. . . . . 6 |- ( w e. ~P z -> w C_ z )
17		imass2 5501	. . . . . 6 |- ( w C_ z -> ( F " w ) C_ ( F " z ) )
18	15, 16, 17	3syl 18	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> ( F " w ) C_ ( F " z ) )
19		nrmsep3 21159	. . . . 5 |- ( ( (KQ ` J ) e. Nrm /\ ( ( F " z ) e. (KQ ` J ) /\ ( F " w ) e. ( Clsd ` (KQ ` J ) ) /\ ( F " w ) C_ ( F " z ) ) ) -> E. m e. (KQ ` J ) ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) )
20	3, 8, 13, 18, 19	syl13anc 1328	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> E. m e. (KQ ` J ) ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) )
21		simplll 798	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> J e. (TopOn ` X ) )
22	6	kqid 21531	. . . . . . 7 |- ( J e. (TopOn ` X ) -> F e. ( J Cn (KQ ` J ) ) )
23	21, 22	syl 17	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> F e. ( J Cn (KQ ` J ) ) )
24		simprl 794	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> m e. (KQ ` J ) )
25		cnima 21069	. . . . . 6 |- ( ( F e. ( J Cn (KQ ` J ) ) /\ m e. (KQ ` J ) ) -> ( `' F " m ) e. J )
26	23, 24, 25	syl2anc 693	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( `' F " m ) e. J )
27		simprrl 804	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( F " w ) C_ m )
28	6	kqffn 21528	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> F Fn X )
29		fnfun 5988	. . . . . . . 8 |- ( F Fn X -> Fun F )
30	21, 28, 29	3syl 18	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> Fun F )
31	11	adantr 481	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> w e. ( Clsd ` J ) )
32		eqid 2622	. . . . . . . . . 10 |- U. J = U. J
33	32	cldss 20833	. . . . . . . . 9 |- ( w e. ( Clsd ` J ) -> w C_ U. J )
34	31, 33	syl 17	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> w C_ U. J )
35		fndm 5990	. . . . . . . . . 10 |- ( F Fn X -> dom F = X )
36	21, 28, 35	3syl 18	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> dom F = X )
37		toponuni 20719	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> X = U. J )
38	21, 37	syl 17	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> X = U. J )
39	36, 38	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> dom F = U. J )
40	34, 39	sseqtr4d 3642	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> w C_ dom F )
41		funimass3 6333	. . . . . . 7 |- ( ( Fun F /\ w C_ dom F ) -> ( ( F " w ) C_ m <-> w C_ ( `' F " m ) ) )
42	30, 40, 41	syl2anc 693	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( ( F " w ) C_ m <-> w C_ ( `' F " m ) ) )
43	27, 42	mpbid 222	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> w C_ ( `' F " m ) )
44	6	kqtopon 21530	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> (KQ ` J ) e. (TopOn ` ran F ) )
45		topontop 20718	. . . . . . . . . 10 |- ( (KQ ` J ) e. (TopOn ` ran F ) -> (KQ ` J ) e. Top )
46	21, 44, 45	3syl 18	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> (KQ ` J ) e. Top )
47		elssuni 4467	. . . . . . . . . 10 |- ( m e. (KQ ` J ) -> m C_ U. (KQ ` J ) )
48	47	ad2antrl 764	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> m C_ U. (KQ ` J ) )
49		eqid 2622	. . . . . . . . . 10 |- U. (KQ ` J ) = U. (KQ ` J )
50	49	clscld 20851	. . . . . . . . 9 |- ( ( (KQ ` J ) e. Top /\ m C_ U. (KQ ` J ) ) -> ( ( cls ` (KQ ` J ) ) ` m ) e. ( Clsd ` (KQ ` J ) ) )
51	46, 48, 50	syl2anc 693	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( ( cls ` (KQ ` J ) ) ` m ) e. ( Clsd ` (KQ ` J ) ) )
52		cnclima 21072	. . . . . . . 8 |- ( ( F e. ( J Cn (KQ ` J ) ) /\ ( ( cls ` (KQ ` J ) ) ` m ) e. ( Clsd ` (KQ ` J ) ) ) -> ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) e. ( Clsd ` J ) )
53	23, 51, 52	syl2anc 693	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) e. ( Clsd ` J ) )
54	49	sscls 20860	. . . . . . . . 9 |- ( ( (KQ ` J ) e. Top /\ m C_ U. (KQ ` J ) ) -> m C_ ( ( cls ` (KQ ` J ) ) ` m ) )
55	46, 48, 54	syl2anc 693	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> m C_ ( ( cls ` (KQ ` J ) ) ` m ) )
56		imass2 5501	. . . . . . . 8 |- ( m C_ ( ( cls ` (KQ ` J ) ) ` m ) -> ( `' F " m ) C_ ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) )
57	55, 56	syl 17	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( `' F " m ) C_ ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) )
58	32	clsss2 20876	. . . . . . 7 |- ( ( ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) e. ( Clsd ` J ) /\ ( `' F " m ) C_ ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) ) -> ( ( cls ` J ) ` ( `' F " m ) ) C_ ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) )
59	53, 57, 58	syl2anc 693	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( ( cls ` J ) ` ( `' F " m ) ) C_ ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) )
60		simprrr 805	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) )
61		imass2 5501	. . . . . . . 8 |- ( ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) -> ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) C_ ( `' F " ( F " z ) ) )
62	60, 61	syl 17	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) C_ ( `' F " ( F " z ) ) )
63	5	adantr 481	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> z e. J )
64	6	kqsat 21534	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ z e. J ) -> ( `' F " ( F " z ) ) = z )
65	21, 63, 64	syl2anc 693	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( `' F " ( F " z ) ) = z )
66	62, 65	sseqtrd 3641	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( `' F " ( ( cls ` (KQ ` J ) ) ` m ) ) C_ z )
67	59, 66	sstrd 3613	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> ( ( cls ` J ) ` ( `' F " m ) ) C_ z )
68		sseq2 3627	. . . . . . 7 |- ( u = ( `' F " m ) -> ( w C_ u <-> w C_ ( `' F " m ) ) )
69		fveq2 6191	. . . . . . . 8 |- ( u = ( `' F " m ) -> ( ( cls ` J ) ` u ) = ( ( cls ` J ) ` ( `' F " m ) ) )
70	69	sseq1d 3632	. . . . . . 7 |- ( u = ( `' F " m ) -> ( ( ( cls ` J ) ` u ) C_ z <-> ( ( cls ` J ) ` ( `' F " m ) ) C_ z ) )
71	68, 70	anbi12d 747	. . . . . 6 |- ( u = ( `' F " m ) -> ( ( w C_ u /\ ( ( cls ` J ) ` u ) C_ z ) <-> ( w C_ ( `' F " m ) /\ ( ( cls ` J ) ` ( `' F " m ) ) C_ z ) ) )
72	71	rspcev 3309	. . . . 5 |- ( ( ( `' F " m ) e. J /\ ( w C_ ( `' F " m ) /\ ( ( cls ` J ) ` ( `' F " m ) ) C_ z ) ) -> E. u e. J ( w C_ u /\ ( ( cls ` J ) ` u ) C_ z ) )
73	26, 43, 67, 72	syl12anc 1324	. . . 4 |- ( ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) /\ ( m e. (KQ ` J ) /\ ( ( F " w ) C_ m /\ ( ( cls ` (KQ ` J ) ) ` m ) C_ ( F " z ) ) ) ) -> E. u e. J ( w C_ u /\ ( ( cls ` J ) ` u ) C_ z ) )
74	20, 73	rexlimddv 3035	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) /\ ( z e. J /\ w e. ( ( Clsd ` J ) i^i ~P z ) ) ) -> E. u e. J ( w C_ u /\ ( ( cls ` J ) ` u ) C_ z ) )
75	74	ralrimivva 2971	. 2 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) -> A. z e. J A. w e. ( ( Clsd ` J ) i^i ~P z ) E. u e. J ( w C_ u /\ ( ( cls ` J ) ` u ) C_ z ) )
76		isnrm 21139	. 2 |- ( J e. Nrm <-> ( J e. Top /\ A. z e. J A. w e. ( ( Clsd ` J ) i^i ~P z ) E. u e. J ( w C_ u /\ ( ( cls ` J ) ` u ) C_ z ) ) )
77	2, 75, 76	sylanbrc 698	1 |- ( ( J e. (TopOn ` X ) /\ (KQ ` J ) e. Nrm ) -> J e. Nrm )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Clsdccld 20820   clsccl 20822   Cn ccn 21028   Nrmcnrm 21114  KQckq 21496

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-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-int 4476  df-iun 4522  df-iin 4523  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-map 7859  df-qtop 16167  df-top 20699  df-topon 20716  df-cld 20823  df-cls 20825  df-cn 21031  df-nrm 21121  df-kq 21497

This theorem is referenced by:  kqnrm  21555