Theorem tngval 22443

Description: Value of the function which augments a given structure G with a norm N. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
tngval.t	|- T = ( G toNrmGrp N )
tngval.m	|- .- = ( -g ` G )
tngval.d	|- D = ( N o. .- )
tngval.j	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
tngval	|- ( ( G e. V /\ N e. W ) -> T = ( ( G sSet <. ( dist ` ndx ) , D >. ) sSet <. (TopSet ` ndx ) , J >. ) )

Proof of Theorem tngval

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tngval.t	. 2 |- T = ( G toNrmGrp N )
2		elex 3212	. . 3 |- ( G e. V -> G e. _V )
3		elex 3212	. . 3 |- ( N e. W -> N e. _V )
4		simpl 473	. . . . . 6 |- ( ( g = G /\ f = N ) -> g = G )
5		simpr 477	. . . . . . . . 9 |- ( ( g = G /\ f = N ) -> f = N )
6	4	fveq2d 6195	. . . . . . . . . 10 |- ( ( g = G /\ f = N ) -> ( -g ` g ) = ( -g ` G ) )
7		tngval.m	. . . . . . . . . 10 |- .- = ( -g ` G )
8	6, 7	syl6eqr 2674	. . . . . . . . 9 |- ( ( g = G /\ f = N ) -> ( -g ` g ) = .- )
9	5, 8	coeq12d 5286	. . . . . . . 8 |- ( ( g = G /\ f = N ) -> ( f o. ( -g ` g ) ) = ( N o. .- ) )
10		tngval.d	. . . . . . . 8 |- D = ( N o. .- )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( ( g = G /\ f = N ) -> ( f o. ( -g ` g ) ) = D )
12	11	opeq2d 4409	. . . . . 6 |- ( ( g = G /\ f = N ) -> <. ( dist ` ndx ) , ( f o. ( -g ` g ) ) >. = <. ( dist ` ndx ) , D >. )
13	4, 12	oveq12d 6668	. . . . 5 |- ( ( g = G /\ f = N ) -> ( g sSet <. ( dist ` ndx ) , ( f o. ( -g ` g ) ) >. ) = ( G sSet <. ( dist ` ndx ) , D >. ) )
14	11	fveq2d 6195	. . . . . . 7 |- ( ( g = G /\ f = N ) -> ( MetOpen ` ( f o. ( -g ` g ) ) ) = ( MetOpen ` D ) )
15		tngval.j	. . . . . . 7 |- J = ( MetOpen ` D )
16	14, 15	syl6eqr 2674	. . . . . 6 |- ( ( g = G /\ f = N ) -> ( MetOpen ` ( f o. ( -g ` g ) ) ) = J )
17	16	opeq2d 4409	. . . . 5 |- ( ( g = G /\ f = N ) -> <. (TopSet ` ndx ) , ( MetOpen ` ( f o. ( -g ` g ) ) ) >. = <. (TopSet ` ndx ) , J >. )
18	13, 17	oveq12d 6668	. . . 4 |- ( ( g = G /\ f = N ) -> ( ( g sSet <. ( dist ` ndx ) , ( f o. ( -g ` g ) ) >. ) sSet <. (TopSet ` ndx ) , ( MetOpen ` ( f o. ( -g ` g ) ) ) >. ) = ( ( G sSet <. ( dist ` ndx ) , D >. ) sSet <. (TopSet ` ndx ) , J >. ) )
19		df-tng 22389	. . . 4 |- toNrmGrp = ( g e. _V , f e. _V |-> ( ( g sSet <. ( dist ` ndx ) , ( f o. ( -g ` g ) ) >. ) sSet <. (TopSet ` ndx ) , ( MetOpen ` ( f o. ( -g ` g ) ) ) >. ) )
20		ovex 6678	. . . 4 |- ( ( G sSet <. ( dist ` ndx ) , D >. ) sSet <. (TopSet ` ndx ) , J >. ) e. _V
21	18, 19, 20	ovmpt2a 6791	. . 3 |- ( ( G e. _V /\ N e. _V ) -> ( G toNrmGrp N ) = ( ( G sSet <. ( dist ` ndx ) , D >. ) sSet <. (TopSet ` ndx ) , J >. ) )
22	2, 3, 21	syl2an 494	. 2 |- ( ( G e. V /\ N e. W ) -> ( G toNrmGrp N ) = ( ( G sSet <. ( dist ` ndx ) , D >. ) sSet <. (TopSet ` ndx ) , J >. ) )
23	1, 22	syl5eq 2668	1 |- ( ( G e. V /\ N e. W ) -> T = ( ( G sSet <. ( dist ` ndx ) , D >. ) sSet <. (TopSet ` ndx ) , J >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   o. ccom 5118   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855  TopSetcts 15947   distcds 15950   -gcsg 17424   MetOpencmopn 19736   toNrmGrp ctng 22383

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-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tng 22389

This theorem is referenced by:  tnglem  22444  tngds  22452  tngtset  22453