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Mirrors > Home > HSE Home > Th. List > hvsubvali | Structured version Visualization version GIF version |
Description: Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddcl.1 | ⊢ 𝐴 ∈ ℋ |
hvaddcl.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvsubvali | ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvaddcl.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | hvaddcl.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hvsubval 27257 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = 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: hvsubsub4i 27300 hvnegdii 27303 hvsubeq0i 27304 hvsubcan2i 27305 hvsubaddi 27307 normlem0 27350 normlem9 27359 norm3difi 27388 normpar2i 27397 pjsubii 27921 pjssmii 27924 pjcji 27927 lnophmlem2 28260 |
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