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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 28968 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2744 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ) |
5 | oveq1 7190 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) | |
6 | 4, 5 | sylbi 220 | . . 3 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) |
7 | neg1cn 11843 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | 7, 2 | hvmulcli 28962 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
9 | 1, 8, 2 | hvadd32i 29002 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) |
10 | 1, 2, 8 | hvassi 29001 | . . . . 5 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
11 | 2 | hvnegidi 28978 | . . . . . . 7 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 11 | oveq2i 7194 | . . . . . 6 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
13 | ax-hvaddid 28952 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
14 | 1, 13 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
15 | 12, 14 | eqtri 2762 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
16 | 10, 15 | eqtri 2762 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = 𝐴 |
17 | 9, 16 | eqtri 2762 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = 𝐴 |
18 | 2 | hvaddid2i 28977 | . . 3 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
19 | 6, 17, 18 | 3eqtr3g 2797 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → 𝐴 = 𝐵) |
20 | oveq1 7190 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = (𝐵 −ℎ 𝐵)) | |
21 | hvsubid 28974 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝐵 −ℎ 𝐵) = 0ℎ) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ (𝐵 −ℎ 𝐵) = 0ℎ |
23 | 20, 22 | eqtrdi 2790 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = 0ℎ) |
24 | 19, 23 | impbii 212 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∈ wcel 2114 (class class class)co 7183 1c1 10629 -cneg 10962 ℋchba 28867 +ℎ cva 28868 ·ℎ csm 28869 0ℎc0v 28872 −ℎ cmv 28873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-hvcom 28949 ax-hvass 28950 ax-hv0cl 28951 ax-hvaddid 28952 ax-hfvmul 28953 ax-hvmulid 28954 ax-hvdistr2 28957 ax-hvmul0 28958 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-ltxr 10771 df-sub 10963 df-neg 10964 df-hvsub 28919 |
This theorem is referenced by: hvsubeq0 29016 bcseqi 29068 normsub0i 29083 pjss2i 29628 |
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