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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 28797 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2826 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ) |
5 | oveq1 7163 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) | |
6 | 4, 5 | sylbi 219 | . . 3 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) |
7 | neg1cn 11752 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | 7, 2 | hvmulcli 28791 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
9 | 1, 8, 2 | hvadd32i 28831 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) |
10 | 1, 2, 8 | hvassi 28830 | . . . . 5 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
11 | 2 | hvnegidi 28807 | . . . . . . 7 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 11 | oveq2i 7167 | . . . . . 6 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
13 | ax-hvaddid 28781 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
14 | 1, 13 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
15 | 12, 14 | eqtri 2844 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
16 | 10, 15 | eqtri 2844 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = 𝐴 |
17 | 9, 16 | eqtri 2844 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = 𝐴 |
18 | 2 | hvaddid2i 28806 | . . 3 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
19 | 6, 17, 18 | 3eqtr3g 2879 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → 𝐴 = 𝐵) |
20 | oveq1 7163 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = (𝐵 −ℎ 𝐵)) | |
21 | hvsubid 28803 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝐵 −ℎ 𝐵) = 0ℎ) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ (𝐵 −ℎ 𝐵) = 0ℎ |
23 | 20, 22 | syl6eq 2872 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = 0ℎ) |
24 | 19, 23 | impbii 211 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 1c1 10538 -cneg 10871 ℋchba 28696 +ℎ cva 28697 ·ℎ csm 28698 0ℎc0v 28701 −ℎ cmv 28702 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvdistr2 28786 ax-hvmul0 28787 |
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 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-hvsub 28748 |
This theorem is referenced by: hvsubeq0 28845 bcseqi 28897 normsub0i 28912 pjss2i 29457 |
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