Theorem nmvs 22480

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 22479	. . . 4 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
8	7	simprbi 480	. . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))
9		oveq1 6657	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
10	9	fveq2d 6195	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦)))
11		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
12	11	oveq1d 6665	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))
13	10, 12	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))))
14		oveq2 6658	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
15	14	fveq2d 6195	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌)))
16		fveq2 6191	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌))
17	16	oveq2d 6666	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))
18	15, 17	eqeq12d 2637	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))))
19	13, 18	rspc2v 3322	. . 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 1262	1 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650   · cmul 9941  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  LModclmod 18863  normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384  NrmModcnlm 22385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-nlm 22391

This theorem is referenced by:  nlmdsdi  22485  nlmdsdir  22486  nlmmul0or  22487  lssnlm  22505  nmoleub2lem3  22915  nmoleub3  22919  ncvsprp  22952  cphnmvs  22990  nmmulg  30012