![]() |
Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > subrval | Structured version Visualization version GIF version |
Description: Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
subrval | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3487 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3487 | . 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
3 | fveq1 6883 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑣) = (𝐴‘𝑣)) | |
4 | fveq1 6883 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑣) = (𝐵‘𝑣)) | |
5 | 3, 4 | oveqan12d 7423 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥‘𝑣) − (𝑦‘𝑣)) = ((𝐴‘𝑣) − (𝐵‘𝑣))) |
6 | 5 | mpteq2dv 5243 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣)))) |
7 | df-subr 43780 | . . 3 ⊢ -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣)))) | |
8 | reex 11200 | . . . 4 ⊢ ℝ ∈ V | |
9 | 8 | mptex 7219 | . . 3 ⊢ (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣))) ∈ V |
10 | 6, 7, 9 | ovmpoa 7558 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣)))) |
11 | 1, 2, 10 | syl2an 595 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 − cmin 11445 -𝑟cminusr 43774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-cnex 11165 ax-resscn 11166 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-subr 43780 |
This theorem is referenced by: subrfv 43786 subrfn 43789 |
Copyright terms: Public domain | W3C validator |