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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 27261 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2615 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ) |
5 | oveq1 6556 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) | |
6 | 4, 5 | sylbi 206 | . . 3 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) |
7 | neg1cn 11001 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | 7, 2 | hvmulcli 27255 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
9 | 1, 8, 2 | hvadd32i 27295 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) |
10 | 1, 2, 8 | hvassi 27294 | . . . . 5 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
11 | 2 | hvnegidi 27271 | . . . . . . 7 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 11 | oveq2i 6560 | . . . . . 6 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
13 | ax-hvaddid 27245 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
14 | 1, 13 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
15 | 12, 14 | eqtri 2632 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
16 | 10, 15 | eqtri 2632 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = 𝐴 |
17 | 9, 16 | eqtri 2632 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = 𝐴 |
18 | 2 | hvaddid2i 27270 | . . 3 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
19 | 6, 17, 18 | 3eqtr3g 2667 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → 𝐴 = 𝐵) |
20 | oveq1 6556 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = (𝐵 −ℎ 𝐵)) | |
21 | hvsubid 27267 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝐵 −ℎ 𝐵) = 0ℎ) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ (𝐵 −ℎ 𝐵) = 0ℎ |
23 | 20, 22 | syl6eq 2660 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = 0ℎ) |
24 | 19, 23 | impbii 198 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 1c1 9816 -cneg 10146 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 0ℎc0v 27165 −ℎ cmv 27166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvdistr2 27250 ax-hvmul0 27251 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-hvsub 27212 |
This theorem is referenced by: hvsubeq0 27309 bcseqi 27361 normsub0i 27376 pjss2i 27923 |
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