h1de2bi - Hilbert Space Explorer

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Theorem h1de2bi 28985

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 28528	. . . 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 3021	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 ↔ 𝐵 ≠ 0_ℎ)
5		h1de2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
6	5, 1	h1de2i 28984	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
7	6	adantl 475	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
8	7	oveq2d 6938	. . . . . 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 28510	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
10	9	recclzi 11100	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (1 / (𝐵 ·_ih 𝐵)) ∈ ℂ)
11		ax-hvmulass 28436	. . . . . . . . . . 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 1526	. . . . . . . . . 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 10330	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
15	14, 9	divcan1zi 11111	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) = 1)
16	15	oveq1d 6937	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = (1 ·_ℎ 𝐴))
17	13, 16	eqtr3d 2816	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = (1 ·_ℎ 𝐴))
18		ax-hvmulid 28435	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
19	5, 18	ax-mp 5	. . . . . . . 8 ⊢ (1 ·_ℎ 𝐴) = 𝐴
20	17, 19	syl6eq 2830	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
21	20	adantr 474	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
22	8, 21	eqtr3d 2816	. . . . 5 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = 𝐴)
23	5, 1	hicli 28510	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
24		ax-hvmulass 28436	. . . . . . . . 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 1526	. . . . . . . 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 10358	. . . . . . . . . 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 580	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
29	23, 9	divreczi 11113	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
30	28, 29	eqtr4d 2817	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)))
31	30	oveq1d 6937	. . . . . . 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 2816	. . . . . 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 474	. . . . 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 2816	. . . 4 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
35	34	ex 403	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
36	23, 9	divclzi 11110	. . . . 5 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ)
37	1	elexi 3415	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
38	37	snss 4549	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ ↔ {𝐵} ⊆ ℋ)
39	1, 38	mpbi 222	. . . . . . . . 9 ⊢ {𝐵} ⊆ ℋ
40		occl 28735	. . . . . . . . 9 ⊢ ({𝐵} ⊆ ℋ → (⊥‘{𝐵}) ∈ C_ℋ )
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ (⊥‘{𝐵}) ∈ C_ℋ
42	41	choccli 28738	. . . . . . 7 ⊢ (⊥‘(⊥‘{𝐵})) ∈ C_ℋ
43	42	chshii 28656	. . . . . 6 ⊢ (⊥‘(⊥‘{𝐵})) ∈ S_ℋ
44		h1did 28982	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (⊥‘(⊥‘{𝐵})))
45	1, 44	ax-mp 5	. . . . . 6 ⊢ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))
46		shmulcl 28647	. . . . . 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 1525	. . . . 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 2847	. . . 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 239	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
51	35, 50	impbid 204	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
52	4, 51	sylbir 227	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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 {csn 4398 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 · cmul 10277 / cdiv 11032 ℋchba 28348 ·_ℎ csm 28350 ·_ih csp 28351 0_ℎc0v 28353 S_ℋ csh 28357 C_ℋ cch 28358 ⊥cort 28359

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvmulass 28436 ax-hvdistr1 28437 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his2 28512 ax-his3 28513 ax-his4 28514 ax-hcompl 28631

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cn 21439 df-cnp 21440 df-lm 21441 df-haus 21527 df-tx 21774 df-hmeo 21967 df-xms 22533 df-ms 22534 df-tms 22535 df-cau 23462 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 df-dip 28128 df-hnorm 28397 df-hvsub 28400 df-hlim 28401 df-hcau 28402 df-sh 28636 df-ch 28650 df-oc 28681

This theorem is referenced by: h1de2ctlem 28986 elspansn2 28998

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