Theorem nmvs 22290

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 22289	. . . 4 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
8	7	simprbi 479	. . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))
9		oveq1 6556	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
10	9	fveq2d 6107	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
11		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
12	11	oveq1d 6564	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))
13	10, 12	eqeq12d 2625	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))))
14		oveq2 6557	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
15	14	fveq2d 6107	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
16		fveq2 6103	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌))
17	16	oveq2d 6565	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))
18	15, 17	eqeq12d 2625	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
19	13, 18	rspc2v 3293	. . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
20	8, 19	syl5com 31	. 2 ⊢ (𝑊 ∈ NrmMod → ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
21	20	3impib 1254	1 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549   · cmul 9820  Basecbs 15695  Scalarcsca 15771   ·_𝑠 cvsca 15772  LModclmod 18686  normcnm 22191  NrmGrpcngp 22192  NrmRingcnrg 22194  NrmModcnlm 22195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-nlm 22201

This theorem is referenced by:  nlmdsdi  22295  nlmdsdir  22296  nlmmul0or  22297  lssnlm  22315  nmoleub2lem3  22723  nmoleub3  22727  ncvsprp  22760  cphnmvs  22798  nmmulg  29340