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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 30956 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | oveq2i 7437 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
5 | neg1cn 12380 | . . . . . 6 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 30950 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 1, 2, 1, 6 | hvadd4i 30994 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((𝐴 +ℎ 𝐴) +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
8 | hv2times 30997 | . . . . . . 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 2735 | . . . . 5 ⊢ (𝐴 +ℎ 𝐴) = (2 ·ℎ 𝐴) |
11 | 2 | hvnegidi 30966 | . . . . 5 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 10, 11 | oveq12i 7438 | . . . 4 ⊢ ((𝐴 +ℎ 𝐴) +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = ((2 ·ℎ 𝐴) +ℎ 0ℎ) |
13 | 7, 12 | eqtri 2754 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((2 ·ℎ 𝐴) +ℎ 0ℎ) |
14 | 2cn 12341 | . . . . 5 ⊢ 2 ∈ ℂ | |
15 | 14, 1 | hvmulcli 30950 | . . . 4 ⊢ (2 ·ℎ 𝐴) ∈ ℋ |
16 | ax-hvaddid 30940 | . . . 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 2754 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (2 ·ℎ 𝐴) |
19 | 4, 18 | eqtri 2754 | 1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = (2 ·ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7426 1c1 11161 -cneg 11497 2c2 12321 ℋchba 30855 +ℎ cva 30856 ·ℎ csm 30857 0ℎc0v 30860 −ℎ cmv 30861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-hfvadd 30936 ax-hvcom 30937 ax-hvass 30938 ax-hvaddid 30940 ax-hfvmul 30941 ax-hvmulid 30942 ax-hvdistr1 30944 ax-hvdistr2 30945 ax-hvmul0 30946 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-ltxr 11305 df-sub 11498 df-neg 11499 df-2 12329 df-hvsub 30907 |
This theorem is referenced by: normpar2i 31092 |
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