h1de2bi - Hilbert Space Explorer

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Theorem h1de2bi 29323

Description: Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
h1de2.1	⊢ 𝐴 ∈ ℋ
h1de2.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
h1de2bi	⊢ (𝐵 ≠ 0_ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))

**Proof of Theorem h1de2bi**
Step	Hyp	Ref	Expression
1		h1de2.2	. . . 4 ⊢ 𝐵 ∈ ℋ
2		his6 28868	. . . 4 ⊢ (𝐵 ∈ ℋ → ((𝐵 ·_ih 𝐵) = 0 ↔ 𝐵 = 0_ℎ))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐵 ·_ih 𝐵) = 0 ↔ 𝐵 = 0_ℎ)
4	3	necon3bii 3066	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 ↔ 𝐵 ≠ 0_ℎ)
5		h1de2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
6	5, 1	h1de2i 29322	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
7	6	adantl 484	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
8	7	oveq2d 7164	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)))
9	1, 1	hicli 28850	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
10	9	recclzi 11357	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (1 / (𝐵 ·_ih 𝐵)) ∈ ℂ)
11		ax-hvmulass 28776	. . . . . . . . . . 11 ⊢ (((1 / (𝐵 ·_ih 𝐵)) ∈ ℂ ∧ (𝐵 ·_ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)))
12	9, 5, 11	mp3an23 1446	. . . . . . . . . 10 ⊢ ((1 / (𝐵 ·_ih 𝐵)) ∈ ℂ → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)))
13	10, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)))
14		ax-1cn 10587	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
15	14, 9	divcan1zi 11368	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) = 1)
16	15	oveq1d 7163	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = (1 ·_ℎ 𝐴))
17	13, 16	eqtr3d 2856	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = (1 ·_ℎ 𝐴))
18		ax-hvmulid 28775	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
19	5, 18	ax-mp 5	. . . . . . . 8 ⊢ (1 ·_ℎ 𝐴) = 𝐴
20	17, 19	syl6eq 2870	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
21	20	adantr 483	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
22	8, 21	eqtr3d 2856	. . . . 5 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = 𝐴)
23	5, 1	hicli 28850	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
24		ax-hvmulass 28776	. . . . . . . . 9 ⊢ (((1 / (𝐵 ·_ih 𝐵)) ∈ ℂ ∧ (𝐴 ·_ih 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℋ) → (((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) ·_ℎ 𝐵) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)))
25	23, 1, 24	mp3an23 1446	. . . . . . . 8 ⊢ ((1 / (𝐵 ·_ih 𝐵)) ∈ ℂ → (((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) ·_ℎ 𝐵) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)))
26	10, 25	syl 17	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) ·_ℎ 𝐵) = ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)))
27		mulcom 10615	. . . . . . . . . 10 ⊢ (((1 / (𝐵 ·_ih 𝐵)) ∈ ℂ ∧ (𝐴 ·_ih 𝐵) ∈ ℂ) → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
28	10, 23, 27	sylancl 588	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
29	23, 9	divreczi 11370	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
30	28, 29	eqtr4d 2857	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)))
31	30	oveq1d 7163	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) ·_ℎ 𝐵) = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
32	26, 31	eqtr3d 2856	. . . . . 6 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
33	32	adantr 483	. . . . 5 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
34	22, 33	eqtr3d 2856	. . . 4 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
35	34	ex 415	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
36	23, 9	divclzi 11367	. . . . 5 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ)
37	1	elexi 3512	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
38	37	snss 4710	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ ↔ {𝐵} ⊆ ℋ)
39	1, 38	mpbi 232	. . . . . . . . 9 ⊢ {𝐵} ⊆ ℋ
40		occl 29073	. . . . . . . . 9 ⊢ ({𝐵} ⊆ ℋ → (⊥‘{𝐵}) ∈ C_ℋ )
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ (⊥‘{𝐵}) ∈ C_ℋ
42	41	choccli 29076	. . . . . . 7 ⊢ (⊥‘(⊥‘{𝐵})) ∈ C_ℋ
43	42	chshii 28996	. . . . . 6 ⊢ (⊥‘(⊥‘{𝐵})) ∈ S_ℋ
44		h1did 29320	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (⊥‘(⊥‘{𝐵})))
45	1, 44	ax-mp 5	. . . . . 6 ⊢ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))
46		shmulcl 28987	. . . . . 6 ⊢ (((⊥‘(⊥‘{𝐵})) ∈ S_ℋ ∧ ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ ∧ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))) → (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵})))
47	43, 45, 46	mp3an13 1445	. . . . 5 ⊢ (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ → (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵})))
48	36, 47	syl 17	. . . 4 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵})))
49		eleq1 2898	. . . 4 ⊢ (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵}))))
50	48, 49	syl5ibrcom 249	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
51	35, 50	impbid 214	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
52	4, 51	sylbir 237	1 ⊢ (𝐵 ≠ 0_ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1530 ∈ wcel 2107 ≠ wne 3014 ⊆ wss 3934 {csn 4559 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 · cmul 10534 / cdiv 11289 ℋchba 28688 ·_ℎ csm 28690 ·_ih csp 28691 0_ℎc0v 28693 S_ℋ csh 28697 C_ℋ cch 28698 ⊥cort 28699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1904 ax-6 1963 ax-7 2008 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2153 ax-12 2169 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 ax-hilex 28768 ax-hfvadd 28769 ax-hvcom 28770 ax-hvass 28771 ax-hv0cl 28772 ax-hvaddid 28773 ax-hfvmul 28774 ax-hvmulid 28775 ax-hvmulass 28776 ax-hvdistr1 28777 ax-hvdistr2 28778 ax-hvmul0 28779 ax-hfi 28848 ax-his1 28851 ax-his2 28852 ax-his3 28853 ax-his4 28854 ax-hcompl 28971

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1082 df-3an 1083 df-tru 1533 df-fal 1543 df-ex 1774 df-nf 1778 df-sb 2063 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-cnp 21828 df-lm 21829 df-haus 21915 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-cau 23851 df-grpo 28262 df-gid 28263 df-ginv 28264 df-gdiv 28265 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-vs 28368 df-nmcv 28369 df-ims 28370 df-dip 28470 df-hnorm 28737 df-hvsub 28740 df-hlim 28741 df-hcau 28742 df-sh 28976 df-ch 28990 df-oc 29021

This theorem is referenced by: h1de2ctlem 29324 elspansn2 29336

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