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Mirrors > Home > HSE Home > Th. List > hvsubcan | Structured version Visualization version GIF version |
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubcan | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = (𝐴 −ℎ 𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvsubval 28001 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
2 | 1 | 3adant3 1101 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
3 | hvsubval 28001 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶))) | |
4 | 3 | 3adant2 1100 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶))) |
5 | 2, 4 | eqeq12d 2666 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = (𝐴 −ℎ 𝐶) ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
6 | neg1cn 11162 | . . . 4 ⊢ -1 ∈ ℂ | |
7 | hvmulcl 27998 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
8 | 6, 7 | mpan 706 | . . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
9 | hvmulcl 27998 | . . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ 𝐶) ∈ ℋ) | |
10 | 6, 9 | mpan 706 | . . . 4 ⊢ (𝐶 ∈ ℋ → (-1 ·ℎ 𝐶) ∈ ℋ) |
11 | hvaddcan 28055 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ ∧ (-1 ·ℎ 𝐶) ∈ ℋ) → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) ↔ (-1 ·ℎ 𝐵) = (-1 ·ℎ 𝐶))) | |
12 | 10, 11 | syl3an3 1401 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) ↔ (-1 ·ℎ 𝐵) = (-1 ·ℎ 𝐶))) |
13 | 8, 12 | syl3an2 1400 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) ↔ (-1 ·ℎ 𝐵) = (-1 ·ℎ 𝐶))) |
14 | neg1ne0 11164 | . . . . 5 ⊢ -1 ≠ 0 | |
15 | 6, 14 | pm3.2i 470 | . . . 4 ⊢ (-1 ∈ ℂ ∧ -1 ≠ 0) |
16 | hvmulcan 28057 | . . . 4 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) = (-1 ·ℎ 𝐶) ↔ 𝐵 = 𝐶)) | |
17 | 15, 16 | mp3an1 1451 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) = (-1 ·ℎ 𝐶) ↔ 𝐵 = 𝐶)) |
18 | 17 | 3adant1 1099 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) = (-1 ·ℎ 𝐶) ↔ 𝐵 = 𝐶)) |
19 | 5, 13, 18 | 3bitrd 294 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = (𝐴 −ℎ 𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 (class class class)co 6690 ℂcc 9972 0cc0 9974 1c1 9975 -cneg 10305 ℋchil 27904 +ℎ cva 27905 ·ℎ csm 27906 −ℎ cmv 27910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-hfvadd 27985 ax-hvcom 27986 ax-hvass 27987 ax-hv0cl 27988 ax-hvaddid 27989 ax-hfvmul 27990 ax-hvmulid 27991 ax-hvmulass 27992 ax-hvdistr1 27993 ax-hvdistr2 27994 ax-hvmul0 27995 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-hvsub 27956 |
This theorem is referenced by: hvsubcan2 28060 |
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