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Mirrors > Home > HSE Home > Th. List > hi2eq | Structured version Visualization version GIF version |
Description: Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hi2eq | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvsubcl 30950 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | |
2 | his2sub 31025 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (𝐴 −ℎ 𝐵) ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = ((𝐴 ·ih (𝐴 −ℎ 𝐵)) − (𝐵 ·ih (𝐴 −ℎ 𝐵)))) | |
3 | 1, 2 | mpd3an3 1459 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = ((𝐴 ·ih (𝐴 −ℎ 𝐵)) − (𝐵 ·ih (𝐴 −ℎ 𝐵)))) |
4 | 3 | eqeq1d 2728 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = 0 ↔ ((𝐴 ·ih (𝐴 −ℎ 𝐵)) − (𝐵 ·ih (𝐴 −ℎ 𝐵))) = 0)) |
5 | his6 31032 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) ∈ ℋ → (((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = 0 ↔ (𝐴 −ℎ 𝐵) = 0ℎ)) | |
6 | 1, 5 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = 0 ↔ (𝐴 −ℎ 𝐵) = 0ℎ)) |
7 | 4, 6 | bitr3d 280 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 ·ih (𝐴 −ℎ 𝐵)) − (𝐵 ·ih (𝐴 −ℎ 𝐵))) = 0 ↔ (𝐴 −ℎ 𝐵) = 0ℎ)) |
8 | hicl 31013 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐴 −ℎ 𝐵) ∈ ℋ) → (𝐴 ·ih (𝐴 −ℎ 𝐵)) ∈ ℂ) | |
9 | 1, 8 | syldan 589 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝐴 −ℎ 𝐵)) ∈ ℂ) |
10 | simpr 483 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 𝐵 ∈ ℋ) | |
11 | hicl 31013 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 −ℎ 𝐵) ∈ ℋ) → (𝐵 ·ih (𝐴 −ℎ 𝐵)) ∈ ℂ) | |
12 | 10, 1, 11 | syl2anc 582 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ·ih (𝐴 −ℎ 𝐵)) ∈ ℂ) |
13 | 9, 12 | subeq0ad 11631 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 ·ih (𝐴 −ℎ 𝐵)) − (𝐵 ·ih (𝐴 −ℎ 𝐵))) = 0 ↔ (𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)))) |
14 | hvsubeq0 31001 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵)) | |
15 | 7, 13, 14 | 3bitr3d 308 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 (class class class)co 7424 ℂcc 11156 0cc0 11158 − cmin 11494 ℋchba 30852 ·ih csp 30855 0ℎc0v 30857 −ℎ cmv 30858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-hfvadd 30933 ax-hvcom 30934 ax-hvass 30935 ax-hv0cl 30936 ax-hvaddid 30937 ax-hfvmul 30938 ax-hvmulid 30939 ax-hvdistr2 30942 ax-hvmul0 30943 ax-hfi 31012 ax-his2 31016 ax-his3 31017 ax-his4 31018 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-neg 11497 df-hvsub 30904 |
This theorem is referenced by: hial2eq 31039 |
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