Theorem ismntoplly 30069

Description: Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)

Assertion

Ref	Expression
ismntoplly	|- ( ( N e. NN0 /\ J e. V ) -> ( NManTop J <-> ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) ) )

Proof of Theorem ismntoplly

Dummy variables j n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( n = N /\ j = J ) -> n = N )
2	1	eleq1d 2686	. . . 4 |- ( ( n = N /\ j = J ) -> ( n e. NN0 <-> N e. NN0 ) )
3		simpr 477	. . . . . 6 |- ( ( n = N /\ j = J ) -> j = J )
4	3	eleq1d 2686	. . . . 5 |- ( ( n = N /\ j = J ) -> ( j e. 2ndc <-> J e. 2ndc ) )
5	3	eleq1d 2686	. . . . 5 |- ( ( n = N /\ j = J ) -> ( j e. Haus <-> J e. Haus ) )
6		fveq2 6191	. . . . . . . . . 10 |- ( n = N -> (𝔼hil ` n ) = (𝔼hil ` N ) )
7	6	fveq2d 6195	. . . . . . . . 9 |- ( n = N -> ( TopOpen ` (𝔼hil ` n ) ) = ( TopOpen ` (𝔼hil ` N ) ) )
8	7	eceq1d 7783	. . . . . . . 8 |- ( n = N -> [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= = [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= )
9		llyeq 21273	. . . . . . . 8 |- ( [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= = [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= -> Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= = Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= )
10	8, 9	syl 17	. . . . . . 7 |- ( n = N -> Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= = Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= )
11	10	adantr 481	. . . . . 6 |- ( ( n = N /\ j = J ) -> Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= = Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= )
12	3, 11	eleq12d 2695	. . . . 5 |- ( ( n = N /\ j = J ) -> ( j e. Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= <-> J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) )
13	4, 5, 12	3anbi123d 1399	. . . 4 |- ( ( n = N /\ j = J ) -> ( ( j e. 2ndc /\ j e. Haus /\ j e. Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= ) <-> ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) ) )
14	2, 13	anbi12d 747	. . 3 |- ( ( n = N /\ j = J ) -> ( ( n e. NN0 /\ ( j e. 2ndc /\ j e. Haus /\ j e. Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= ) ) <-> ( N e. NN0 /\ ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) ) ) )
15		df-mntop 30067	. . 3 |- ManTop = { <. n , j >. | ( n e. NN0 /\ ( j e. 2ndc /\ j e. Haus /\ j e. Locally [ ( TopOpen ` (𝔼hil ` n ) ) ] ~= ) ) }
16	14, 15	brabga 4989	. 2 |- ( ( N e. NN0 /\ J e. V ) -> ( NManTop J <-> ( N e. NN0 /\ ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) ) ) )
17		simpl 473	. . 3 |- ( ( N e. NN0 /\ J e. V ) -> N e. NN0 )
18	17	biantrurd 529	. 2 |- ( ( N e. NN0 /\ J e. V ) -> ( ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) <-> ( N e. NN0 /\ ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) ) ) )
19	16, 18	bitr4d 271	1 |- ( ( N e. NN0 /\ J e. V ) -> ( NManTop J <-> ( J e. 2ndc /\ J e. Haus /\ J e. Locally [ ( TopOpen ` (𝔼hil ` N ) ) ] ~= ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   [cec 7740   NN0cn0 11292   TopOpenctopn 16082   Hauscha 21112   2ndcc2ndc 21241  Locally clly 21267   ~= chmph 21557  𝔼_hilcehl 23172  ManTopcmntop 30066

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ec 7744  df-lly 21269  df-mntop 30067

This theorem is referenced by:  ismntop  30070