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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 30888 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
2 | 1 | oveq1d 7434 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴)) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
3 | ax-1cn 11198 | . . . . 5 ⊢ 1 ∈ ℂ | |
4 | neg1cn 12359 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | ax-hvdistr2 30891 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) | |
6 | 3, 4, 5 | mp3an12 1447 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) |
7 | hvsubval 30898 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) | |
8 | 7 | anidms 565 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
9 | 2, 6, 8 | 3eqtr4rd 2776 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = ((1 + -1) ·ℎ 𝐴)) |
10 | 1pneg1e0 12364 | . . . 4 ⊢ (1 + -1) = 0 | |
11 | 10 | oveq1i 7429 | . . 3 ⊢ ((1 + -1) ·ℎ 𝐴) = (0 ·ℎ 𝐴) |
12 | 9, 11 | eqtrdi 2781 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (0 ·ℎ 𝐴)) |
13 | ax-hvmul0 30892 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | |
14 | 12, 13 | eqtrd 2765 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 -cneg 11477 ℋchba 30801 +ℎ cva 30802 ·ℎ csm 30803 0ℎc0v 30806 −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-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-hvmulid 30888 ax-hvdistr2 30891 ax-hvmul0 30892 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-neg 11479 df-hvsub 30853 |
This theorem is referenced by: hvnegid 30909 hvsubeq0i 30945 hvaddsub4 30960 norm3difi 31029 5oalem1 31536 5oalem2 31537 5oalem3 31538 5oalem5 31540 3oalem2 31545 pjsslem 31561 ho0val 31632 lnop0 31848 0cnop 31861 pjclem4 32081 pj3si 32089 |
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