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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 28721 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 −ℎ 𝐴) ∈ ℋ) | |
2 | 1 | 3adant3 1124 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 −ℎ 𝐴) ∈ ℋ) |
3 | hvsubcl 28721 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 −ℎ 𝐵) ∈ ℋ) | |
4 | 3 | 3adant2 1123 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 −ℎ 𝐵) ∈ ℋ) |
5 | neg1cn 11739 | . . . . . 6 ⊢ -1 ∈ ℂ | |
6 | neg1ne0 11741 | . . . . . 6 ⊢ -1 ≠ 0 | |
7 | 5, 6 | pm3.2i 471 | . . . . 5 ⊢ (-1 ∈ ℂ ∧ -1 ≠ 0) |
8 | hvmulcan 28776 | . . . . 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 1439 | . . . 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 28771 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐴)) = (𝐴 −ℎ 𝐶)) | |
12 | 11 | 3adant3 1124 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐴)) = (𝐴 −ℎ 𝐶)) |
13 | hvnegdi 28771 | . . . . 5 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐵)) = (𝐵 −ℎ 𝐶)) | |
14 | 13 | 3adant2 1123 | . . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ (𝐶 −ℎ 𝐵)) = (𝐵 −ℎ 𝐶)) |
15 | 12, 14 | eqeq12d 2834 | . . 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 28778 | . . 3 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 −ℎ 𝐴) = (𝐶 −ℎ 𝐵) ↔ 𝐴 = 𝐵)) | |
17 | 10, 15, 16 | 3bitr3d 310 | . 2 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = (𝐵 −ℎ 𝐶) ↔ 𝐴 = 𝐵)) |
18 | 17 | 3coml 1119 | 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 -cneg 10859 ℋchba 28623 ·ℎ csm 28625 −ℎ cmv 28629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvmulass 28711 ax-hvdistr1 28712 ax-hvdistr2 28713 ax-hvmul0 28714 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-hvsub 28675 |
This theorem is referenced by: hvaddsub4 28782 |
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