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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 22526 | . . . 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 6697 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
10 | 9 | fveq2d 6233 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦))) |
11 | fveq2 6229 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋)) | |
12 | 11 | oveq1d 6705 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))) |
13 | 10, 12 | eqeq12d 2666 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))) |
14 | oveq2 6698 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
15 | 14 | fveq2d 6233 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌))) |
16 | fveq2 6229 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌)) | |
17 | 16 | oveq2d 6706 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) |
18 | 15, 17 | eqeq12d 2666 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))) |
19 | 13, 18 | rspc2v 3353 | . . 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 1281 | 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 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ‘cfv 5926 (class class class)co 6690 · cmul 9979 Basecbs 15904 Scalarcsca 15991 ·𝑠 cvsca 15992 LModclmod 18911 normcnm 22428 NrmGrpcngp 22429 NrmRingcnrg 22431 NrmModcnlm 22432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-iota 5889 df-fv 5934 df-ov 6693 df-nlm 22438 |
This theorem is referenced by: nlmdsdi 22532 nlmdsdir 22533 nlmmul0or 22534 lssnlm 22552 nmoleub2lem3 22961 nmoleub3 22965 ncvsprp 22998 cphnmvs 23036 nmmulg 30140 |
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