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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 7426 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝑦))) | |
2 | oveq2 7427 | . . 3 ⊢ (𝑦 = 𝐵 → (-1 ·ℎ 𝑦) = (-1 ·ℎ 𝐵)) | |
3 | 2 | oveq2d 7435 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
4 | df-hvsub 30853 | . 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦))) | |
5 | ovex 7452 | . 2 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ V | |
6 | 1, 3, 4, 5 | ovmpo 7581 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 1c1 11141 -cneg 11477 ℋchba 30801 +ℎ cva 30802 ·ℎ csm 30803 −ℎ cmv 30807 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-hvsub 30853 |
This theorem is referenced by: hvsubcl 30899 hvsubvali 30902 hvsubid 30908 hvnegid 30909 hv2neg 30910 hvaddsubval 30915 hvsub4 30919 hvaddsub12 30920 hvpncan 30921 hvaddsubass 30923 hvsubass 30926 hvsubdistr1 30931 hvsubdistr2 30932 hvsubcan 30956 hvsub0 30958 his2sub 30974 hhph 31060 shsubcl 31102 shsel3 31197 honegsubi 31678 lnopsubi 31856 lnfnsubi 31928 superpos 32236 cdj1i 32315 |
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