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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 6556 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝑦))) | |
2 | oveq2 6557 | . . 3 ⊢ (𝑦 = 𝐵 → (-1 ·ℎ 𝑦) = (-1 ·ℎ 𝐵)) | |
3 | 2 | oveq2d 6565 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
4 | df-hvsub 27212 | . 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦))) | |
5 | ovex 6577 | . 2 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ V | |
6 | 1, 3, 4, 5 | ovmpt2 6694 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 1c1 9816 -cneg 10146 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 −ℎ cmv 27166 |
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-sep 4709 ax-nul 4717 ax-pr 4833 |
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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-hvsub 27212 |
This theorem is referenced by: hvsubcl 27258 hvsubvali 27261 hvsubid 27267 hvnegid 27268 hv2neg 27269 hvaddsubval 27274 hvsub4 27278 hvaddsub12 27279 hvpncan 27280 hvaddsubass 27282 hvsubass 27285 hvsubdistr1 27290 hvsubdistr2 27291 hvsubcan 27315 hvsub0 27317 his2sub 27333 hhph 27419 shsubcl 27461 shsel3 27558 honegsubi 28039 lnopsubi 28217 lnfnsubi 28289 superpos 28597 cdj1i 28676 |
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