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Mirrors > Home > MPE Home > Th. List > Mathboxes > subrfn | Structured version Visualization version GIF version |
Description: Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
subrfn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) Fn ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7178 | . . 3 ⊢ ((𝐴‘𝑥) − (𝐵‘𝑥)) ∈ V | |
2 | eqid 2818 | . . 3 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) − (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) − (𝐵‘𝑥))) | |
3 | 1, 2 | fnmpti 6484 | . 2 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) − (𝐵‘𝑥))) Fn ℝ |
4 | subrval 40676 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) − (𝐵‘𝑥)))) | |
5 | 4 | fneq1d 6439 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴-𝑟𝐵) Fn ℝ ↔ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) − (𝐵‘𝑥))) Fn ℝ)) |
6 | 3, 5 | mpbiri 259 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) Fn ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ↦ cmpt 5137 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 − cmin 10858 -𝑟cminusr 40667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-subr 40673 |
This theorem is referenced by: (None) |
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