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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 2823 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | 1, 2, 3, 4, 5 | scaffval 19654 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) |
7 | ovex 7191 | . 2 ⊢ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V | |
8 | 6, 7 | fnmpoi 7770 | 1 ⊢ ∙ Fn (𝐾 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 ·sf cscaf 19637 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-scaf 19639 |
This theorem is referenced by: (None) |
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