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Theorem nmvs 23212
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 23211 . . . 4 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
87simprbi 497 . . 3 (𝑊 ∈ NrmMod → ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
9 fvoveq1 7168 . . . . 5 (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
10 fveq2 6663 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
1110oveq1d 7160 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝑥) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑦)))
129, 11eqeq12d 2834 . . . 4 (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦))))
13 oveq2 7153 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413fveq2d 6667 . . . . 5 (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
15 fveq2 6663 . . . . . 6 (𝑦 = 𝑌 → (𝑁𝑦) = (𝑁𝑌))
1615oveq2d 7161 . . . . 5 (𝑦 = 𝑌 → ((𝐴𝑋) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑌)))
1714, 16eqeq12d 2834 . . . 4 (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
1812, 17rspc2v 3630 . . 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 1108 1 ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145   · cmul 10530  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  LModclmod 19563  normcnm 23113  NrmGrpcngp 23114  NrmRingcnrg 23116  NrmModcnlm 23117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-nlm 23123
This theorem is referenced by:  nlmdsdi  23217  nlmdsdir  23218  nlmmul0or  23219  lssnlm  23237  nmoleub2lem3  23646  nmoleub3  23650  ncvsprp  23683  cphnmvs  23721  nmmulg  31108
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