Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tngval Structured version   Unicode version

Theorem tngval 18685
 Description: Value of the function which augments a given structure with a norm . (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
tngval.t toNrmGrp
tngval.m
tngval.d
tngval.j
Assertion
Ref Expression
tngval sSet sSet TopSet

Proof of Theorem tngval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngval.t . 2 toNrmGrp
2 elex 2966 . . 3
3 elex 2966 . . 3
4 simpl 445 . . . . . 6
5 simpr 449 . . . . . . . . 9
64fveq2d 5735 . . . . . . . . . 10
7 tngval.m . . . . . . . . . 10
86, 7syl6eqr 2488 . . . . . . . . 9
95, 8coeq12d 5040 . . . . . . . 8
10 tngval.d . . . . . . . 8
119, 10syl6eqr 2488 . . . . . . 7
1211opeq2d 3993 . . . . . 6
134, 12oveq12d 6102 . . . . 5 sSet sSet
1411fveq2d 5735 . . . . . . 7
15 tngval.j . . . . . . 7
1614, 15syl6eqr 2488 . . . . . 6
1716opeq2d 3993 . . . . 5 TopSet TopSet
1813, 17oveq12d 6102 . . . 4 sSet sSet TopSet sSet sSet TopSet
19 df-tng 18637 . . . 4 toNrmGrp sSet sSet TopSet
20 ovex 6109 . . . 4 sSet sSet TopSet
2118, 19, 20ovmpt2a 6207 . . 3 toNrmGrp sSet sSet TopSet
222, 3, 21syl2an 465 . 2 toNrmGrp sSet sSet TopSet
231, 22syl5eq 2482 1 sSet sSet TopSet
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cvv 2958  cop 3819   ccom 4885  cfv 5457  (class class class)co 6084  cnx 13471   sSet csts 13472  TopSetcts 13540  cds 13543  csg 14693  cmopn 16696   toNrmGrp ctng 18631 This theorem is referenced by:  tnglem  18686  tngds  18694  tngtset  18695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tng 18637
 Copyright terms: Public domain W3C validator