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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 28720 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 1c1 10526 -cneg 10859 ℋchba 28623 +ℎ cva 28624 ·ℎ csm 28625 −ℎ cmv 28629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-hvsub 28675 |
This theorem is referenced by: hvsubsub4i 28763 hvnegdii 28766 hvsubeq0i 28767 hvsubcan2i 28768 hvsubaddi 28770 normlem0 28813 normlem9 28822 norm3difi 28851 normpar2i 28860 pjsubii 29382 pjssmii 29385 pjcji 29388 lnophmlem2 29721 |
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