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Mirrors > Home > HSE Home > Th. List > hvsub0 | Structured version Visualization version GIF version |
Description: Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsub0 | ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 29038 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | hvsubval 29051 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 −ℎ 0ℎ) = (𝐴 +ℎ (-1 ·ℎ 0ℎ))) | |
3 | 1, 2 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = (𝐴 +ℎ (-1 ·ℎ 0ℎ))) |
4 | neg1cn 11909 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | hvmul0 29059 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1 ·ℎ 0ℎ) = 0ℎ) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (-1 ·ℎ 0ℎ) = 0ℎ |
7 | 6 | oveq2i 7202 | . . 3 ⊢ (𝐴 +ℎ (-1 ·ℎ 0ℎ)) = (𝐴 +ℎ 0ℎ) |
8 | 3, 7 | eqtrdi 2787 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = (𝐴 +ℎ 0ℎ)) |
9 | ax-hvaddid 29039 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
10 | 8, 9 | eqtrd 2771 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 1c1 10695 -cneg 11028 ℋchba 28954 +ℎ cva 28955 ·ℎ csm 28956 0ℎc0v 28959 −ℎ cmv 28960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-hv0cl 29038 ax-hvaddid 29039 ax-hvmulass 29042 ax-hvmul0 29045 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 df-hvsub 29006 |
This theorem is referenced by: normneg 29179 5oalem3 29691 5oalem5 29693 nmcopexi 30062 nmcfnexi 30086 |
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