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Mirrors > Home > HSE Home > Th. List > hvaddcani | Structured version Visualization version GIF version |
Description: Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
hvaddcan.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvaddcani | ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6911 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) → ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐴)) = ((𝐴 +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐴))) | |
2 | hvnegdi.1 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
3 | hvnegdi.2 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
4 | neg1cn 11471 | . . . . . 6 ⊢ -1 ∈ ℂ | |
5 | 4, 2 | hvmulcli 28425 | . . . . 5 ⊢ (-1 ·ℎ 𝐴) ∈ ℋ |
6 | 2, 3, 5 | hvadd32i 28465 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐴)) = ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐵) |
7 | 2 | hvnegidi 28441 | . . . . 5 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐴)) = 0ℎ |
8 | 7 | oveq1i 6914 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵) |
9 | 3 | hvaddid2i 28440 | . . . 4 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
10 | 6, 8, 9 | 3eqtri 2852 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐴)) = 𝐵 |
11 | hvaddcan.3 | . . . . 5 ⊢ 𝐶 ∈ ℋ | |
12 | 2, 11, 5 | hvadd32i 28465 | . . . 4 ⊢ ((𝐴 +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐴)) = ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐶) |
13 | 7 | oveq1i 6914 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐶) = (0ℎ +ℎ 𝐶) |
14 | 11 | hvaddid2i 28440 | . . . 4 ⊢ (0ℎ +ℎ 𝐶) = 𝐶 |
15 | 12, 13, 14 | 3eqtri 2852 | . . 3 ⊢ ((𝐴 +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐴)) = 𝐶 |
16 | 1, 10, 15 | 3eqtr3g 2883 | . 2 ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) → 𝐵 = 𝐶) |
17 | oveq2 6912 | . 2 ⊢ (𝐵 = 𝐶 → (𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶)) | |
18 | 16, 17 | impbii 201 | 1 ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∈ wcel 2166 (class class class)co 6904 1c1 10252 -cneg 10585 ℋchba 28330 +ℎ cva 28331 ·ℎ csm 28332 0ℎc0v 28335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-hvcom 28412 ax-hvass 28413 ax-hv0cl 28414 ax-hvaddid 28415 ax-hfvmul 28416 ax-hvmulid 28417 ax-hvdistr2 28420 ax-hvmul0 28421 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-ltxr 10395 df-sub 10586 df-neg 10587 df-hvsub 28382 |
This theorem is referenced by: hvsubaddi 28477 hvaddcan 28481 |
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