Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > addrval | Structured version Visualization version GIF version |
Description: Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
addrval | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3185 | . 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
3 | fveq1 6102 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑣) = (𝐴‘𝑣)) | |
4 | fveq1 6102 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑣) = (𝐵‘𝑣)) | |
5 | 3, 4 | oveqan12d 6568 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥‘𝑣) + (𝑦‘𝑣)) = ((𝐴‘𝑣) + (𝐵‘𝑣))) |
6 | 5 | mpteq2dv 4673 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
7 | df-addr 37688 | . . 3 ⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣)))) | |
8 | reex 9906 | . . . 4 ⊢ ℝ ∈ V | |
9 | 8 | mptex 6390 | . . 3 ⊢ (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))) ∈ V |
10 | 6, 7, 9 | ovmpt2a 6689 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
11 | 1, 2, 10 | syl2an 493 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 + caddc 9818 +𝑟cplusr 37682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-addr 37688 |
This theorem is referenced by: addrfv 37694 addrfn 37697 |
Copyright terms: Public domain | W3C validator |