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Mirrors > Home > MPE Home > Th. List > nlmnrg | Structured version Visualization version GIF version |
Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmnrg.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
nlmnrg | ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | nlmnrg.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | eqid 2821 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2821 | . . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 23284 | . . 3 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (((norm‘𝐹)‘𝑥) · ((norm‘𝑊)‘𝑦)))) |
8 | 7 | simplbi 500 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing)) |
9 | 8 | simp3d 1140 | 1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 · cmul 10542 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 LModclmod 19634 normcnm 23186 NrmGrpcngp 23187 NrmRingcnrg 23189 NrmModcnlm 23190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-nlm 23196 |
This theorem is referenced by: nlmngp2 23289 nlmtlm 23303 nvctvc 23309 lssnlm 23310 |
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