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Theorem isnlm 18703
 Description: A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnlm.v
isnlm.n
isnlm.s
isnlm.f Scalar
isnlm.k
isnlm.a
Assertion
Ref Expression
isnlm NrmMod NrmGrp NrmRing
Distinct variable groups:   ,,   ,,   ,,   ,   ,,   , ,
Allowed substitution hints:   (,)   ()

Proof of Theorem isnlm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anass 631 . 2 NrmGrp NrmRing NrmGrp NrmRing
2 df-3an 938 . . . 4 NrmGrp NrmRing NrmGrp NrmRing
3 elin 3522 . . . . 5 NrmGrp NrmGrp
43anbi1i 677 . . . 4 NrmGrp NrmRing NrmGrp NrmRing
52, 4bitr4i 244 . . 3 NrmGrp NrmRing NrmGrp NrmRing
65anbi1i 677 . 2 NrmGrp NrmRing NrmGrp NrmRing
7 fvex 5734 . . . . 5 Scalar
87a1i 11 . . . 4 Scalar
9 id 20 . . . . . . 7 Scalar Scalar
10 fveq2 5720 . . . . . . . 8 Scalar Scalar
11 isnlm.f . . . . . . . 8 Scalar
1210, 11syl6eqr 2485 . . . . . . 7 Scalar
139, 12sylan9eqr 2489 . . . . . 6 Scalar
1413eleq1d 2501 . . . . 5 Scalar NrmRing NrmRing
1513fveq2d 5724 . . . . . . 7 Scalar
16 isnlm.k . . . . . . 7
1715, 16syl6eqr 2485 . . . . . 6 Scalar
18 simpl 444 . . . . . . . . 9 Scalar
1918fveq2d 5724 . . . . . . . 8 Scalar
20 isnlm.v . . . . . . . 8
2119, 20syl6eqr 2485 . . . . . . 7 Scalar
2218fveq2d 5724 . . . . . . . . . 10 Scalar
23 isnlm.n . . . . . . . . . 10
2422, 23syl6eqr 2485 . . . . . . . . 9 Scalar
2518fveq2d 5724 . . . . . . . . . . 11 Scalar
26 isnlm.s . . . . . . . . . . 11
2725, 26syl6eqr 2485 . . . . . . . . . 10 Scalar
2827oveqd 6090 . . . . . . . . 9 Scalar
2924, 28fveq12d 5726 . . . . . . . 8 Scalar
3013fveq2d 5724 . . . . . . . . . . 11 Scalar
31 isnlm.a . . . . . . . . . . 11
3230, 31syl6eqr 2485 . . . . . . . . . 10 Scalar
3332fveq1d 5722 . . . . . . . . 9 Scalar
3424fveq1d 5722 . . . . . . . . 9 Scalar
3533, 34oveq12d 6091 . . . . . . . 8 Scalar
3629, 35eqeq12d 2449 . . . . . . 7 Scalar
3721, 36raleqbidv 2908 . . . . . 6 Scalar
3817, 37raleqbidv 2908 . . . . 5 Scalar
3914, 38anbi12d 692 . . . 4 Scalar NrmRing NrmRing
408, 39sbcied 3189 . . 3 Scalar NrmRing NrmRing
41 df-nlm 18626 . . 3 NrmMod NrmGrp Scalar NrmRing
4240, 41elrab2 3086 . 2 NrmMod NrmGrp NrmRing
431, 6, 423bitr4ri 270 1 NrmMod NrmGrp NrmRing
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  cvv 2948  wsbc 3153   cin 3311  cfv 5446  (class class class)co 6073   cmul 8987  cbs 13461  Scalarcsca 13524  cvsca 13525  clmod 15942  cnm 18616  NrmGrpcngp 18617  NrmRingcnrg 18619  NrmModcnlm 18620 This theorem is referenced by:  nmvs  18704  nlmngp  18705  nlmlmod  18706  nlmnrg  18707  sranlm  18712  lssnlm  18728  tchcph  19186  cnzh  24346  rezh  24347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlm 18626
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