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Mirrors > Home > MPE Home > Th. List > nmvs | Structured version Visualization version GIF version |
Description: Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
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) |
Ref | Expression |
---|---|
nmvs | ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) |
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 24653 | . . . 4 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))) |
8 | 7 | simprbi 495 | . . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))) |
9 | fvoveq1 7442 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦))) | |
10 | fveq2 6896 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋)) | |
11 | 10 | oveq1d 7434 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))) |
12 | 9, 11 | eqeq12d 2741 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))) |
13 | oveq2 7427 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
14 | 13 | fveq2d 6900 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌))) |
15 | fveq2 6896 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌)) | |
16 | 15 | oveq2d 7435 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) |
17 | 14, 16 | eqeq12d 2741 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))) |
18 | 12, 17 | rspc2v 3617 | . . 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 31 | . 2 ⊢ (𝑊 ∈ NrmMod → ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))) |
20 | 19 | 3impib 1113 | 1 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ‘cfv 6549 (class class class)co 7419 · cmul 11150 Basecbs 17199 Scalarcsca 17255 ·𝑠 cvsca 17256 LModclmod 20772 normcnm 24546 NrmGrpcngp 24547 NrmRingcnrg 24549 NrmModcnlm 24550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-nlm 24556 |
This theorem is referenced by: nlmdsdi 24659 nlmdsdir 24660 nlmmul0or 24661 lssnlm 24679 nmoleub2lem3 25103 nmoleub3 25107 ncvsprp 25141 cphnmvs 25179 nmmulg 33720 |
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