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Mirrors > Home > MPE Home > Th. List > scafval | Structured version Visualization version GIF version |
Description: The scalar multiplication operation as 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) |
scaffval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
scafval | ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = (𝑋 · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7222 | . 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 · 𝑦) = (𝑋 · 𝑌)) | |
2 | scaffval.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | scaffval.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | scaffval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
5 | scaffval.a | . . 3 ⊢ ∙ = ( ·sf ‘𝑊) | |
6 | scaffval.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | 2, 3, 4, 5, 6 | scaffval 19917 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) |
8 | ovex 7246 | . 2 ⊢ (𝑋 · 𝑌) ∈ V | |
9 | 1, 7, 8 | ovmpoa 7364 | 1 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = (𝑋 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 Scalarcsca 16805 ·𝑠 cvsca 16806 ·sf cscaf 19900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-scaf 19902 |
This theorem is referenced by: lmodfopne 19937 cnmpt1vsca 23091 cnmpt2vsca 23092 nlmvscn 23585 |
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