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Mirrors > Home > MPE Home > Th. List > lmodscaf | Structured version Visualization version GIF version |
Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
scaffval.b | ⊢ 𝐵 = (Base‘𝑊) |
scaffval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
scaffval.k | ⊢ 𝐾 = (Base‘𝐹) |
scaffval.a | ⊢ ∙ = ( ·sf ‘𝑊) |
Ref | Expression |
---|---|
lmodscaf | ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaffval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
2 | scaffval.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2821 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | scaffval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
5 | 1, 2, 3, 4 | lmodvscl 19645 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵) |
6 | 5 | 3expb 1116 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵) |
7 | 6 | ralrimivva 3191 | . 2 ⊢ (𝑊 ∈ LMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵) |
8 | scaffval.a | . . . 4 ⊢ ∙ = ( ·sf ‘𝑊) | |
9 | 1, 2, 4, 8, 3 | scaffval 19646 | . . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) |
10 | 9 | fmpo 7760 | . 2 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵 ↔ ∙ :(𝐾 × 𝐵)⟶𝐵) |
11 | 7, 10 | sylib 220 | 1 ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 LModclmod 19628 ·sf cscaf 19629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-lmod 19630 df-scaf 19631 |
This theorem is referenced by: lmodfopnelem1 19664 nlmvscn 23290 cvsi 23728 |
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