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Mirrors > Home > HSE Home > Th. List > hvsubid | Structured version Visualization version GIF version |
Description: Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubid | ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hvmulid 28203 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
2 | 1 | oveq1d 6808 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴)) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
3 | ax-1cn 10196 | . . . . 5 ⊢ 1 ∈ ℂ | |
4 | neg1cn 11326 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | ax-hvdistr2 28206 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) | |
6 | 3, 4, 5 | mp3an12 1562 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) |
7 | hvsubval 28213 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) | |
8 | 7 | anidms 556 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
9 | 2, 6, 8 | 3eqtr4rd 2816 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = ((1 + -1) ·ℎ 𝐴)) |
10 | 1pneg1e0 11331 | . . . 4 ⊢ (1 + -1) = 0 | |
11 | 10 | oveq1i 6803 | . . 3 ⊢ ((1 + -1) ·ℎ 𝐴) = (0 ·ℎ 𝐴) |
12 | 9, 11 | syl6eq 2821 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (0 ·ℎ 𝐴)) |
13 | ax-hvmul0 28207 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | |
14 | 12, 13 | eqtrd 2805 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 (class class class)co 6793 ℂcc 10136 0cc0 10138 1c1 10139 + caddc 10141 -cneg 10469 ℋchil 28116 +ℎ cva 28117 ·ℎ csm 28118 0ℎc0v 28121 −ℎ cmv 28122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-hvmulid 28203 ax-hvdistr2 28206 ax-hvmul0 28207 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 df-sub 10470 df-neg 10471 df-hvsub 28168 |
This theorem is referenced by: hvnegid 28224 hvsubeq0i 28260 hvaddsub4 28275 norm3difi 28344 5oalem1 28853 5oalem2 28854 5oalem3 28855 5oalem5 28857 3oalem2 28862 pjsslem 28878 ho0val 28949 lnop0 29165 0cnop 29178 pjclem4 29398 pj3si 29406 |
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