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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulvfn | Structured version Visualization version GIF version |
Description: Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
mulvfn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) Fn ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7183 | . . 3 ⊢ (𝐴 · (𝐵‘𝑥)) ∈ V | |
2 | eqid 2821 | . . 3 ⊢ (𝑥 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑥))) | |
3 | 1, 2 | fnmpti 6485 | . 2 ⊢ (𝑥 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑥))) Fn ℝ |
4 | mulvval 40793 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) = (𝑥 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑥)))) | |
5 | 4 | fneq1d 6440 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴.𝑣𝐵) Fn ℝ ↔ (𝑥 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑥))) Fn ℝ)) |
6 | 3, 5 | mpbiri 260 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) Fn ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ↦ cmpt 5138 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 · cmul 10536 .𝑣ctimesr 40784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-mulv 40790 |
This theorem is referenced by: (None) |
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