hi2eq - Hilbert Space Explorer

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Theorem hi2eq 30928

Description: Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hi2eq	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih (𝐴 −_ℎ 𝐵)) = (𝐵 ·_ih (𝐴 −_ℎ 𝐵)) ↔ 𝐴 = 𝐵))

**Proof of Theorem hi2eq**
Step	Hyp	Ref	Expression
1		hvsubcl 30840	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) ∈ ℋ)
2		his2sub 30915	. . . . 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 2730	. . 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 30922	. . . 4 ⊢ ((𝐴 −_ℎ 𝐵) ∈ ℋ → (((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = 0 ↔ (𝐴 −_ℎ 𝐵) = 0_ℎ))
6	1, 5	syl 17	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = 0 ↔ (𝐴 −_ℎ 𝐵) = 0_ℎ))
7	4, 6	bitr3d 281	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝐴 ·_ih (𝐴 −_ℎ 𝐵)) − (𝐵 ·_ih (𝐴 −_ℎ 𝐵))) = 0 ↔ (𝐴 −_ℎ 𝐵) = 0_ℎ))
8		hicl 30903	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐴 −_ℎ 𝐵) ∈ ℋ) → (𝐴 ·_ih (𝐴 −_ℎ 𝐵)) ∈ ℂ)
9	1, 8	syldan 590	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝐴 −_ℎ 𝐵)) ∈ ℂ)
10		simpr 484	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → 𝐵 ∈ ℋ)
11		hicl 30903	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 −_ℎ 𝐵) ∈ ℋ) → (𝐵 ·_ih (𝐴 −_ℎ 𝐵)) ∈ ℂ)
12	10, 1, 11	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ·_ih (𝐴 −_ℎ 𝐵)) ∈ ℂ)
13	9, 12	subeq0ad 11612	. 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 30891	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) = 0_ℎ ↔ 𝐴 = 𝐵))
15	7, 13, 14	3bitr3d 309	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 395 = wceq 1534 ∈ wcel 2099 (class class class)co 7420 ℂcc 11137 0cc0 11139 − cmin 11475 ℋchba 30742 ·_ih csp 30745 0_ℎc0v 30747 −_ℎ cmv 30748

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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-hfvadd 30823 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvdistr2 30832 ax-hvmul0 30833 ax-hfi 30902 ax-his2 30906 ax-his3 30907 ax-his4 30908

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 df-neg 11478 df-hvsub 30794

This theorem is referenced by: hial2eq 30929

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