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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 22849 | . . . 4 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))) |
8 | 7 | simprbi 492 | . . 3 ⊢ (𝑊 ∈ NrmMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))) |
9 | fvoveq1 6928 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑁‘(𝑥 · 𝑦)) = (𝑁‘(𝑋 · 𝑦))) | |
10 | fveq2 6433 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋)) | |
11 | 10 | oveq1d 6920 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦))) |
12 | 9, 11 | eqeq12d 2840 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)))) |
13 | oveq2 6913 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
14 | 13 | fveq2d 6437 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑁‘(𝑋 · 𝑦)) = (𝑁‘(𝑋 · 𝑌))) |
15 | fveq2 6433 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑁‘𝑦) = (𝑁‘𝑌)) | |
16 | 15 | oveq2d 6921 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐴‘𝑋) · (𝑁‘𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) |
17 | 14, 16 | eqeq12d 2840 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑁‘(𝑋 · 𝑦)) = ((𝐴‘𝑋) · (𝑁‘𝑦)) ↔ (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))) |
18 | 12, 17 | rspc2v 3539 | . . 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 1150 | 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ‘cfv 6123 (class class class)co 6905 · cmul 10257 Basecbs 16222 Scalarcsca 16308 ·𝑠 cvsca 16309 LModclmod 19219 normcnm 22751 NrmGrpcngp 22752 NrmRingcnrg 22754 NrmModcnlm 22755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-nul 5013 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-nlm 22761 |
This theorem is referenced by: nlmdsdi 22855 nlmdsdir 22856 nlmmul0or 22857 lssnlm 22875 nmoleub2lem3 23284 nmoleub3 23288 ncvsprp 23321 cphnmvs 23359 nmmulg 30557 |
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