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Mirrors > Home > HSE Home > Th. List > hvsubcan2 | Structured version Visualization version GIF version |
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubcan2 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = (𝐵 −ℎ 𝐶) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvsubcl 31045 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 −ℎ 𝐴) ∈ ℋ) | |
2 | 1 | 3adant3 1131 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 −ℎ 𝐴) ∈ ℋ) |
3 | hvsubcl 31045 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 −ℎ 𝐵) ∈ ℋ) | |
4 | 3 | 3adant2 1130 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 −ℎ 𝐵) ∈ ℋ) |
5 | neg1cn 12377 | . . . . . 6 ⊢ -1 ∈ ℂ | |
6 | neg1ne0 12379 | . . . . . 6 ⊢ -1 ≠ 0 | |
7 | 5, 6 | pm3.2i 470 | . . . . 5 ⊢ (-1 ∈ ℂ ∧ -1 ≠ 0) |
8 | hvmulcan 31100 | . . . . 5 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝐶 −ℎ 𝐴) ∈ ℋ ∧ (𝐶 −ℎ 𝐵) ∈ ℋ) → ((-1 ·ℎ (𝐶 −ℎ 𝐴)) = (-1 ·ℎ (𝐶 −ℎ 𝐵)) ↔ (𝐶 −ℎ 𝐴) = (𝐶 −ℎ 𝐵))) | |
9 | 7, 8 | mp3an1 1447 | . . . 4 ⊢ (((𝐶 −ℎ 𝐴) ∈ ℋ ∧ (𝐶 −ℎ 𝐵) ∈ ℋ) → ((-1 ·ℎ (𝐶 −ℎ 𝐴)) = (-1 ·ℎ (𝐶 −ℎ 𝐵)) ↔ (𝐶 −ℎ 𝐴) = (𝐶 −ℎ 𝐵))) |
10 | 2, 4, 9 | syl2anc 584 | . . 3 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((-1 ·ℎ (𝐶 −ℎ 𝐴)) = (-1 ·ℎ (𝐶 −ℎ 𝐵)) ↔ (𝐶 −ℎ 𝐴) = (𝐶 −ℎ 𝐵))) |
11 | hvnegdi 31095 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐴)) = (𝐴 −ℎ 𝐶)) | |
12 | 11 | 3adant3 1131 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐴)) = (𝐴 −ℎ 𝐶)) |
13 | hvnegdi 31095 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐵)) = (𝐵 −ℎ 𝐶)) | |
14 | 13 | 3adant2 1130 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐵)) = (𝐵 −ℎ 𝐶)) |
15 | 12, 14 | eqeq12d 2750 | . . 3 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((-1 ·ℎ (𝐶 −ℎ 𝐴)) = (-1 ·ℎ (𝐶 −ℎ 𝐵)) ↔ (𝐴 −ℎ 𝐶) = (𝐵 −ℎ 𝐶))) |
16 | hvsubcan 31102 | . . 3 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 −ℎ 𝐴) = (𝐶 −ℎ 𝐵) ↔ 𝐴 = 𝐵)) | |
17 | 10, 15, 16 | 3bitr3d 309 | . 2 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = (𝐵 −ℎ 𝐶) ↔ 𝐴 = 𝐵)) |
18 | 17 | 3coml 1126 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = (𝐵 −ℎ 𝐶) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 -cneg 11490 ℋchba 30947 ·ℎ csm 30949 −ℎ cmv 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvmulass 31035 ax-hvdistr1 31036 ax-hvdistr2 31037 ax-hvmul0 31038 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-hvsub 30999 |
This theorem is referenced by: hvaddsub4 31106 |
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