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Theorem nmvs 24697
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 24696 . . . 4 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
87simprbi 496 . . 3 (𝑊 ∈ NrmMod → ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
9 fvoveq1 7454 . . . . 5 (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
10 fveq2 6906 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
1110oveq1d 7446 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝑥) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑦)))
129, 11eqeq12d 2753 . . . 4 (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦))))
13 oveq2 7439 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413fveq2d 6910 . . . . 5 (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
15 fveq2 6906 . . . . . 6 (𝑦 = 𝑌 → (𝑁𝑦) = (𝑁𝑌))
1615oveq2d 7447 . . . . 5 (𝑦 = 𝑌 → ((𝐴𝑋) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑌)))
1714, 16eqeq12d 2753 . . . 4 (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
1812, 17rspc2v 3633 . . 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 1117 1 ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431   · cmul 11160  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  LModclmod 20858  normcnm 24589  NrmGrpcngp 24590  NrmRingcnrg 24592  NrmModcnlm 24593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-nlm 24599
This theorem is referenced by:  nlmdsdi  24702  nlmdsdir  24703  nlmmul0or  24704  lssnlm  24722  nmoleub2lem3  25148  nmoleub3  25152  ncvsprp  25186  cphnmvs  25224  nmmulg  33967
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