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Theorem nmvs 23260
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 23259 . . . 4 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
87simprbi 500 . . 3 (𝑊 ∈ NrmMod → ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)))
9 fvoveq1 7153 . . . . 5 (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
10 fveq2 6643 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
1110oveq1d 7145 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝑥) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑦)))
129, 11eqeq12d 2837 . . . 4 (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦))))
13 oveq2 7138 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413fveq2d 6647 . . . . 5 (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
15 fveq2 6643 . . . . . 6 (𝑦 = 𝑌 → (𝑁𝑦) = (𝑁𝑌))
1615oveq2d 7146 . . . . 5 (𝑦 = 𝑌 → ((𝐴𝑋) · (𝑁𝑦)) = ((𝐴𝑋) · (𝑁𝑌)))
1714, 16eqeq12d 2837 . . . 4 (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴𝑋) · (𝑁𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌))))
1812, 17rspc2v 3610 . . 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  cfv 6328  (class class class)co 7130   · cmul 10519  Basecbs 16461  Scalarcsca 16546   ·𝑠 cvsca 16547  LModclmod 19609  normcnm 23161  NrmGrpcngp 23162  NrmRingcnrg 23164  NrmModcnlm 23165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-nlm 23171
This theorem is referenced by:  nlmdsdi  23265  nlmdsdir  23266  nlmmul0or  23267  lssnlm  23285  nmoleub2lem3  23698  nmoleub3  23702  ncvsprp  23735  cphnmvs  23773  nmmulg  31216
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