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Mirrors > Home > HSE Home > Th. List > hvsubaddi | Structured version Visualization version GIF version |
Description: Relationship between vector subtraction and 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 |
---|---|
hvsubaddi | ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 28799 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2828 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶) |
5 | neg1cn 11754 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 28793 | . . . . . 6 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 2, 1, 6 | hvadd12i 28836 | . . . . 5 ⊢ (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
8 | 2 | hvnegidi 28809 | . . . . . 6 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
9 | 8 | oveq2i 7169 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
10 | ax-hvaddid 28783 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
12 | 7, 9, 11 | 3eqtri 2850 | . . . 4 ⊢ (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
13 | 12 | eqeq1i 2828 | . . 3 ⊢ ((𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐵 +ℎ 𝐶) ↔ 𝐴 = (𝐵 +ℎ 𝐶)) |
14 | 1, 6 | hvaddcli 28797 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
15 | hvaddcan.3 | . . . 4 ⊢ 𝐶 ∈ ℋ | |
16 | 2, 14, 15 | hvaddcani 28844 | . . 3 ⊢ ((𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐵 +ℎ 𝐶) ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶) |
17 | eqcom 2830 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) ↔ (𝐵 +ℎ 𝐶) = 𝐴) | |
18 | 13, 16, 17 | 3bitr3i 303 | . 2 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
19 | 4, 18 | bitri 277 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 1c1 10540 -cneg 10873 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 0ℎc0v 28703 −ℎ cmv 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvdistr2 28788 ax-hvmul0 28789 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-hvsub 28750 |
This theorem is referenced by: hvsubadd 28856 omlsilem 29181 |
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