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Theorem nmvs 24903
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 24902 . . . 4 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
87simprbi 503 . . 3 (𝑊 ∈ NrmMod → ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
9 fvoveq1 7437 . . . . 5 (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
10 fveq2 6879 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
1110oveq1d 7429 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝑥) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑦)))
129, 11eqeq12d 2776 . . . 4 (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦))))
13 oveq2 7422 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413fveq2d 6883 . . . . 5 (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
15 fveq2 6879 . . . . . 6 (𝑦 = 𝑌 → (𝑁𝑦) = (𝑁𝑌))
1615oveq2d 7430 . . . . 5 (𝑦 = 𝑌 → ((𝐴𝑋) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑌)))
1714, 16eqeq12d 2776 . . . 4 (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
1812, 17rspc2v 3587 . . 3 ((𝑋𝐾𝑌𝑉) → (∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
198, 18syl5com 32 . 2 (𝑊 ∈ NrmMod → ((𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
20193impib 1134 1 ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wa 401  w3a 1103   = wceq 1570  wcel 2145  wral 3076  cfv 6533  (class class class)co 7414   · cmul 11130  Basecbs 17302  Scalarcsca 17346   ·𝑠 cvsca 17347  LModclmod 21045  normcnm 24803  NrmGrpcngp 24804  NrmRingcnrg 24806  NrmModcnlm 24807
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-nlm 24813
This theorem is used by:  nlmdsdi  24908  nlmdsdir  24909  nlmmul0or  24910  lssnlm  24928  nmoleub2lem3  25344  nmoleub3  25348  ncvsprp  25381  cphnmvs  25419  nmmulg  34477
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