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| Mirrors > Home > HSE Home > Th. List > hvsubval | Structured version Visualization version GIF version | ||
| Description: Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hvsubval | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝑦))) | |
| 2 | oveq2 7439 | . . 3 ⊢ (𝑦 = 𝐵 → (-1 ·ℎ 𝑦) = (-1 ·ℎ 𝐵)) | |
| 3 | 2 | oveq2d 7447 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
| 4 | df-hvsub 30990 | . 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦))) | |
| 5 | ovex 7464 | . 2 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ V | |
| 6 | 1, 3, 4, 5 | ovmpo 7593 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 1c1 11156 -cneg 11493 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 −ℎ cmv 30944 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-hvsub 30990 |
| This theorem is referenced by: hvsubcl 31036 hvsubvali 31039 hvsubid 31045 hvnegid 31046 hv2neg 31047 hvaddsubval 31052 hvsub4 31056 hvaddsub12 31057 hvpncan 31058 hvaddsubass 31060 hvsubass 31063 hvsubdistr1 31068 hvsubdistr2 31069 hvsubcan 31093 hvsub0 31095 his2sub 31111 hhph 31197 shsubcl 31239 shsel3 31334 honegsubi 31815 lnopsubi 31993 lnfnsubi 32065 superpos 32373 cdj1i 32452 |
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