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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulvval | Structured version Visualization version GIF version |
Description: Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
mulvval | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) = (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3448 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3448 | . 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
3 | fveq1 6766 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑣) = (𝐵‘𝑣)) | |
4 | oveq12 7277 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ (𝑦‘𝑣) = (𝐵‘𝑣)) → (𝑥 · (𝑦‘𝑣)) = (𝐴 · (𝐵‘𝑣))) | |
5 | 3, 4 | sylan2 593 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 · (𝑦‘𝑣)) = (𝐴 · (𝐵‘𝑣))) |
6 | 5 | mpteq2dv 5176 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))) = (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣)))) |
7 | df-mulv 42042 | . . 3 ⊢ .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣)))) | |
8 | reex 10950 | . . . 4 ⊢ ℝ ∈ V | |
9 | 8 | mptex 7092 | . . 3 ⊢ (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣))) ∈ V |
10 | 6, 7, 9 | ovmpoa 7419 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴.𝑣𝐵) = (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣)))) |
11 | 1, 2, 10 | syl2an 596 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) = (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3430 ↦ cmpt 5157 ‘cfv 6427 (class class class)co 7268 ℝcr 10858 · cmul 10864 .𝑣ctimesr 42036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pr 5351 ax-cnex 10915 ax-resscn 10916 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-oprab 7272 df-mpo 7273 df-mulv 42042 |
This theorem is referenced by: mulvfv 42048 mulvfn 42051 |
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