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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 28791 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | oveq2i 7161 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
5 | neg1cn 11745 | . . . . . 6 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 28785 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 1, 2, 1, 6 | hvadd4i 28829 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((𝐴 +ℎ 𝐴) +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
8 | hv2times 28832 | . . . . . . 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 2830 | . . . . 5 ⊢ (𝐴 +ℎ 𝐴) = (2 ·ℎ 𝐴) |
11 | 2 | hvnegidi 28801 | . . . . 5 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 10, 11 | oveq12i 7162 | . . . 4 ⊢ ((𝐴 +ℎ 𝐴) +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = ((2 ·ℎ 𝐴) +ℎ 0ℎ) |
13 | 7, 12 | eqtri 2844 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((2 ·ℎ 𝐴) +ℎ 0ℎ) |
14 | 2cn 11706 | . . . . 5 ⊢ 2 ∈ ℂ | |
15 | 14, 1 | hvmulcli 28785 | . . . 4 ⊢ (2 ·ℎ 𝐴) ∈ ℋ |
16 | ax-hvaddid 28775 | . . . 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 2844 | . 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (2 ·ℎ 𝐴) |
19 | 4, 18 | eqtri 2844 | 1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐴 −ℎ 𝐵)) = (2 ·ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7150 1c1 10532 -cneg 10865 2c2 11686 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 0ℎc0v 28695 −ℎ cmv 28696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-2 11694 df-hvsub 28742 |
This theorem is referenced by: normpar2i 28927 |
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