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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 2737 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 4 | scaffval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
| 5 | 1, 2, 3, 4 | lmodvscl 20876 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵) |
| 6 | 5 | 3expb 1121 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵) |
| 7 | 6 | ralrimivva 3202 | . 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 20878 | . . 3 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) |
| 10 | 9 | fmpo 8093 | . 2 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵 ↔ ∙ :(𝐾 × 𝐵)⟶𝐵) |
| 11 | 7, 10 | sylib 218 | 1 ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20858 ·sf cscaf 20859 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-lmod 20860 df-scaf 20861 |
| This theorem is referenced by: lmodfopnelem1 20896 nlmvscn 24708 cvsi 25163 |
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