Theorem nmvs 24995

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	⊢ · = ( ·𝑠 ‘𝑊)
isnlm.f	⊢ 𝐹 = (Scalar‘𝑊)
isnlm.k	⊢ 𝐾 = (Base‘𝐹)
isnlm.a	⊢ 𝐴 = (norm‘𝐹)

Assertion

Ref	Expression
nmvs	⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))

Proof of Theorem nmvs

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnlm.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
2		isnlm.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑊)
3		isnlm.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
4		isnlm.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
5		isnlm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
6		isnlm.a	. . . . 5 ⊢ 𝐴 = (norm‘𝐹)
7	1, 2, 3, 4, 5, 6	isnlm 24994	. . . 4 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
8	7	simprbi 503	. . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))
9		fvoveq1 7443	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
10		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
11	10	oveq1d 7435	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))
12	9, 11	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))))
13		oveq2 7428	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
14	13	fveq2d 6889	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
15		fveq2 6885	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌))
16	15	oveq2d 7436	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))
17	14, 16	eqeq12d 2776	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
18	12, 17	rspc2v 3586	. . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
19	8, 18	syl5com 32	. 2 ⊢ (𝑊 ∈ NrmMod → ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
20	19	3impib 1134	1 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6538  (class class class)co 7420   · cmul 11205  Basecbs 17387  Scalarcsca 17431   ·_𝑠 cvsca 17432  LModclmod 21135  normcnm 24895  NrmGrpcngp 24896  NrmRingcnrg 24898  NrmModcnlm 24899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-nlm 24905

This theorem is used by:  nlmdsdi  25000  nlmdsdir  25001  nlmmul0or  25002  lssnlm  25020  nmoleub2lem3  25436  nmoleub3  25440  ncvsprp  25473  cphnmvs  25511  nmmulg  34598