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Mirrors > Home > HSE Home > Th. List > hvsubeq0i | Structured version Visualization version GIF version |
Description: If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvsubeq0i | ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . . . 6 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . . . 6 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 30706 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2736 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ) |
5 | oveq1 7419 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) | |
6 | 4, 5 | sylbi 216 | . . 3 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) |
7 | neg1cn 12333 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | 7, 2 | hvmulcli 30700 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
9 | 1, 8, 2 | hvadd32i 30740 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) |
10 | 1, 2, 8 | hvassi 30739 | . . . . 5 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
11 | 2 | hvnegidi 30716 | . . . . . . 7 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 11 | oveq2i 7423 | . . . . . 6 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
13 | ax-hvaddid 30690 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
14 | 1, 13 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
15 | 12, 14 | eqtri 2759 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
16 | 10, 15 | eqtri 2759 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = 𝐴 |
17 | 9, 16 | eqtri 2759 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = 𝐴 |
18 | 2 | hvaddlidi 30715 | . . 3 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
19 | 6, 17, 18 | 3eqtr3g 2794 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → 𝐴 = 𝐵) |
20 | oveq1 7419 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = (𝐵 −ℎ 𝐵)) | |
21 | hvsubid 30712 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝐵 −ℎ 𝐵) = 0ℎ) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ (𝐵 −ℎ 𝐵) = 0ℎ |
23 | 20, 22 | eqtrdi 2787 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = 0ℎ) |
24 | 19, 23 | impbii 208 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 (class class class)co 7412 1c1 11117 -cneg 11452 ℋchba 30605 +ℎ cva 30606 ·ℎ csm 30607 0ℎc0v 30610 −ℎ cmv 30611 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-hvcom 30687 ax-hvass 30688 ax-hv0cl 30689 ax-hvaddid 30690 ax-hfvmul 30691 ax-hvmulid 30692 ax-hvdistr2 30695 ax-hvmul0 30696 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-sub 11453 df-neg 11454 df-hvsub 30657 |
This theorem is referenced by: hvsubeq0 30754 bcseqi 30806 normsub0i 30821 pjss2i 31366 |
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