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Mirrors > Home > HSE Home > Th. List > hial2eq | Structured version Visualization version GIF version |
Description: Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hial2eq | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvsubcl 28796 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | |
2 | oveq2 7166 | . . . . . 6 ⊢ (𝑥 = (𝐴 −ℎ 𝐵) → (𝐴 ·ih 𝑥) = (𝐴 ·ih (𝐴 −ℎ 𝐵))) | |
3 | oveq2 7166 | . . . . . 6 ⊢ (𝑥 = (𝐴 −ℎ 𝐵) → (𝐵 ·ih 𝑥) = (𝐵 ·ih (𝐴 −ℎ 𝐵))) | |
4 | 2, 3 | eqeq12d 2839 | . . . . 5 ⊢ (𝑥 = (𝐴 −ℎ 𝐵) → ((𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ (𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)))) |
5 | 4 | rspcv 3620 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) ∈ ℋ → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) → (𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)))) |
6 | 1, 5 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) → (𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)))) |
7 | hi2eq 28884 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)) ↔ 𝐴 = 𝐵)) | |
8 | 6, 7 | sylibd 241 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) → 𝐴 = 𝐵)) |
9 | oveq1 7165 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥)) | |
10 | 9 | ralrimivw 3185 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥)) |
11 | 8, 10 | impbid1 227 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 (class class class)co 7158 ℋchba 28698 ·ih csp 28701 −ℎ cmv 28704 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his2 28862 ax-his3 28863 ax-his4 28864 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-hvsub 28750 |
This theorem is referenced by: hial2eq2 28886 hoeq1 29609 hoeq2 29610 unoplin 29699 hmoplin 29721 pjss2coi 29943 pj3cor1i 29988 |
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