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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 6931 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝑦))) | |
2 | oveq2 6932 | . . 3 ⊢ (𝑦 = 𝐵 → (-1 ·ℎ 𝑦) = (-1 ·ℎ 𝐵)) | |
3 | 2 | oveq2d 6940 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
4 | df-hvsub 28404 | . 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦))) | |
5 | ovex 6956 | . 2 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ V | |
6 | 1, 3, 4, 5 | ovmpt2 7075 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 (class class class)co 6924 1c1 10275 -cneg 10609 ℋchba 28352 +ℎ cva 28353 ·ℎ csm 28354 −ℎ cmv 28358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-hvsub 28404 |
This theorem is referenced by: hvsubcl 28450 hvsubvali 28453 hvsubid 28459 hvnegid 28460 hv2neg 28461 hvaddsubval 28466 hvsub4 28470 hvaddsub12 28471 hvpncan 28472 hvaddsubass 28474 hvsubass 28477 hvsubdistr1 28482 hvsubdistr2 28483 hvsubcan 28507 hvsub0 28509 his2sub 28525 hhph 28611 shsubcl 28653 shsel3 28750 honegsubi 29231 lnopsubi 29409 lnfnsubi 29481 superpos 29789 cdj1i 29868 |
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