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Mirrors > Home > HSE Home > Th. List > hvsubcan2i | Structured version Visualization version GIF version |
Description: Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvsubcan2i | ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = (2 ·ℎ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 28428 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | oveq2i 6921 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
5 | neg1cn 11479 | . . . . . 6 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 28422 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 1, 2, 1, 6 | hvadd4i 28466 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((𝐴 +ℎ 𝐴) +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
8 | hv2times 28469 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴)) | |
9 | 1, 8 | ax-mp 5 | . . . . . 6 ⊢ (2 ·ℎ 𝐴) = (𝐴 +ℎ 𝐴) |
10 | 9 | eqcomi 2834 | . . . . 5 ⊢ (𝐴 +ℎ 𝐴) = (2 ·ℎ 𝐴) |
11 | 2 | hvnegidi 28438 | . . . . 5 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 10, 11 | oveq12i 6922 | . . . 4 ⊢ ((𝐴 +ℎ 𝐴) +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = ((2 ·ℎ 𝐴) +ℎ 0ℎ) |
13 | 7, 12 | eqtri 2849 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((2 ·ℎ 𝐴) +ℎ 0ℎ) |
14 | 2cn 11433 | . . . . 5 ⊢ 2 ∈ ℂ | |
15 | 14, 1 | hvmulcli 28422 | . . . 4 ⊢ (2 ·ℎ 𝐴) ∈ ℋ |
16 | ax-hvaddid 28412 | . . . 4 ⊢ ((2 ·ℎ 𝐴) ∈ ℋ → ((2 ·ℎ 𝐴) +ℎ 0ℎ) = (2 ·ℎ 𝐴)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ ((2 ·ℎ 𝐴) +ℎ 0ℎ) = (2 ·ℎ 𝐴) |
18 | 13, 17 | eqtri 2849 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (2 ·ℎ 𝐴) |
19 | 4, 18 | eqtri 2849 | 1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = (2 ·ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 (class class class)co 6910 1c1 10260 -cneg 10593 2c2 11413 ℋchba 28327 +ℎ cva 28328 ·ℎ csm 28329 0ℎc0v 28332 −ℎ cmv 28333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-hfvadd 28408 ax-hvcom 28409 ax-hvass 28410 ax-hvaddid 28412 ax-hfvmul 28413 ax-hvmulid 28414 ax-hvdistr1 28416 ax-hvdistr2 28417 ax-hvmul0 28418 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-sub 10594 df-neg 10595 df-2 11421 df-hvsub 28379 |
This theorem is referenced by: normpar2i 28564 |
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