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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 28800 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | |
2 | oveq2 7143 | . . . . . 6 ⊢ (𝑥 = (𝐴 −ℎ 𝐵) → (𝐴 ·ih 𝑥) = (𝐴 ·ih (𝐴 −ℎ 𝐵))) | |
3 | oveq2 7143 | . . . . . 6 ⊢ (𝑥 = (𝐴 −ℎ 𝐵) → (𝐵 ·ih 𝑥) = (𝐵 ·ih (𝐴 −ℎ 𝐵))) | |
4 | 2, 3 | eqeq12d 2814 | . . . . 5 ⊢ (𝑥 = (𝐴 −ℎ 𝐵) → ((𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ (𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)))) |
5 | 4 | rspcv 3566 | . . . 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 28888 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)) ↔ 𝐴 = 𝐵)) | |
8 | 6, 7 | sylibd 242 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) → 𝐴 = 𝐵)) |
9 | oveq1 7142 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥)) | |
10 | 9 | ralrimivw 3150 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥)) |
11 | 8, 10 | impbid1 228 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 (class class class)co 7135 ℋchba 28702 ·ih csp 28705 −ℎ cmv 28708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 df-hvsub 28754 |
This theorem is referenced by: hial2eq2 28890 hoeq1 29613 hoeq2 29614 unoplin 29703 hmoplin 29725 pjss2coi 29947 pj3cor1i 29992 |
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