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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 24696 | . . . 4 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))) |
| 8 | 7 | simprbi 496 | . . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))) |
| 9 | fvoveq1 7454 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦))) | |
| 10 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋)) | |
| 11 | 10 | oveq1d 7446 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))) |
| 12 | 9, 11 | eqeq12d 2753 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))) |
| 13 | oveq2 7439 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
| 14 | 13 | fveq2d 6910 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌))) |
| 15 | fveq2 6906 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌)) | |
| 16 | 15 | oveq2d 7447 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) |
| 17 | 14, 16 | eqeq12d 2753 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))) |
| 18 | 12, 17 | rspc2v 3633 | . . 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 1117 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 · cmul 11160 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20858 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 NrmModcnlm 24593 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-nlm 24599 |
| This theorem is referenced by: nlmdsdi 24702 nlmdsdir 24703 nlmmul0or 24704 lssnlm 24722 nmoleub2lem3 25148 nmoleub3 25152 ncvsprp 25186 cphnmvs 25224 nmmulg 33967 |
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