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Theorem tngval 21021
 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 3127 . . 3
3 elex 3127 . . 3
4 simpl 457 . . . . . 6
5 simpr 461 . . . . . . . . 9
64fveq2d 5876 . . . . . . . . . 10
7 tngval.m . . . . . . . . . 10
86, 7syl6eqr 2526 . . . . . . . . 9
95, 8coeq12d 5173 . . . . . . . 8
10 tngval.d . . . . . . . 8
119, 10syl6eqr 2526 . . . . . . 7
1211opeq2d 4226 . . . . . 6
134, 12oveq12d 6313 . . . . 5 sSet sSet
1411fveq2d 5876 . . . . . . 7
15 tngval.j . . . . . . 7
1614, 15syl6eqr 2526 . . . . . 6
1716opeq2d 4226 . . . . 5 TopSet TopSet
1813, 17oveq12d 6313 . . . 4 sSet sSet TopSet sSet sSet TopSet
19 df-tng 20973 . . . 4 toNrmGrp sSet sSet TopSet
20 ovex 6320 . . . 4 sSet sSet TopSet
2118, 19, 20ovmpt2a 6428 . . 3 toNrmGrp sSet sSet TopSet
222, 3, 21syl2an 477 . 2 toNrmGrp sSet sSet TopSet
231, 22syl5eq 2520 1 sSet sSet TopSet
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  cvv 3118  cop 4039   ccom 5009  cfv 5594  (class class class)co 6295  cnx 14504   sSet csts 14505  TopSetcts 14578  cds 14581  csg 15927  cmopn 18278   toNrmGrp ctng 20967 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-tng 20973 This theorem is referenced by:  tnglem  21022  tngds  21030  tngtset  21031
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