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Theorem nmvs 24654
Description: Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnlm.v 𝑉 = (Base‘𝑊)
isnlm.n 𝑁 = (norm‘𝑊)
isnlm.s · = ( ·𝑠𝑊)
isnlm.f 𝐹 = (Scalar‘𝑊)
isnlm.k 𝐾 = (Base‘𝐹)
isnlm.a 𝐴 = (norm‘𝐹)
Assertion
Ref Expression
nmvs ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))

Proof of Theorem nmvs
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlm.v . . . . 5 𝑉 = (Base‘𝑊)
2 isnlm.n . . . . 5 𝑁 = (norm‘𝑊)
3 isnlm.s . . . . 5 · = ( ·𝑠𝑊)
4 isnlm.f . . . . 5 𝐹 = (Scalar‘𝑊)
5 isnlm.k . . . . 5 𝐾 = (Base‘𝐹)
6 isnlm.a . . . . 5 𝐴 = (norm‘𝐹)
71, 2, 3, 4, 5, 6isnlm 24653 . . . 4 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
87simprbi 495 . . 3 (𝑊 ∈ NrmMod → ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
9 fvoveq1 7442 . . . . 5 (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
10 fveq2 6896 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
1110oveq1d 7434 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝑥) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑦)))
129, 11eqeq12d 2741 . . . 4 (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦))))
13 oveq2 7427 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413fveq2d 6900 . . . . 5 (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
15 fveq2 6896 . . . . . 6 (𝑦 = 𝑌 → (𝑁𝑦) = (𝑁𝑌))
1615oveq2d 7435 . . . . 5 (𝑦 = 𝑌 → ((𝐴𝑋) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑌)))
1714, 16eqeq12d 2741 . . . 4 (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
1812, 17rspc2v 3617 . . 3 ((𝑋𝐾𝑌𝑉) → (∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
198, 18syl5com 31 . 2 (𝑊 ∈ NrmMod → ((𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
20193impib 1113 1 ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cfv 6549  (class class class)co 7419   · cmul 11150  Basecbs 17199  Scalarcsca 17255   ·𝑠 cvsca 17256  LModclmod 20772  normcnm 24546  NrmGrpcngp 24547  NrmRingcnrg 24549  NrmModcnlm 24550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-nlm 24556
This theorem is referenced by:  nlmdsdi  24659  nlmdsdir  24660  nlmmul0or  24661  lssnlm  24679  nmoleub2lem3  25103  nmoleub3  25107  ncvsprp  25141  cphnmvs  25179  nmmulg  33720
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