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Mirrors > Home > MPE Home > Th. List > scaffn | 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 |
---|---|
scaffn | ⊢ ∙ Fn (𝐾 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaffval.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | scaffval.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | scaffval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | scaffval.a | . . 3 ⊢ ∙ = ( ·sf ‘𝑊) | |
5 | eqid 2731 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | 1, 2, 3, 4, 5 | scaffval 20439 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) |
7 | ovex 7426 | . 2 ⊢ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V | |
8 | 6, 7 | fnmpoi 8038 | 1 ⊢ ∙ Fn (𝐾 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 × cxp 5667 Fn wfn 6527 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 Scalarcsca 17182 ·𝑠 cvsca 17183 ·sf cscaf 20421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-scaf 20423 |
This theorem is referenced by: (None) |
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