Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > addrfn | Structured version Visualization version GIF version |
Description: Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
addrfn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7340 | . . 3 ⊢ ((𝐴‘𝑥) + (𝐵‘𝑥)) ∈ V | |
2 | eqid 2736 | . . 3 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) | |
3 | 1, 2 | fnmpti 6606 | . 2 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) Fn ℝ |
4 | addrval 42297 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))) | |
5 | 4 | fneq1d 6557 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴+𝑟𝐵) Fn ℝ ↔ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) Fn ℝ)) |
6 | 3, 5 | mpbiri 258 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2104 ↦ cmpt 5164 Fn wfn 6453 ‘cfv 6458 (class class class)co 7307 ℝcr 10920 + caddc 10924 +𝑟cplusr 42288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-cnex 10977 ax-resscn 10978 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-addr 42294 |
This theorem is referenced by: addrcom 42306 |
Copyright terms: Public domain | W3C validator |