h1de2bi - Hilbert Space Explorer

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Theorem h1de2bi 31906

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 31451	. . . 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 3010	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 ↔ 𝐵 ≠ 0_ℎ)
5		h1de2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
6	5, 1	h1de2i 31905	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
7	6	adantl 486	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
8	7	oveq2d 7426	. . . . . 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 31433	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
10	9	recclzi 11935	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (1 / (𝐵 ·_ih 𝐵)) ∈ ℂ)
11		ax-hvmulass 31359	. . . . . . . . . . 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 1482	. . . . . . . . . 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 18	. . . . . . . . 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 11153	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
15	14, 9	divcan1zi 11946	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) = 1)
16	15	oveq1d 7425	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = (1 ·_ℎ 𝐴))
17	13, 16	eqtr3d 2800	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = (1 ·_ℎ 𝐴))
18		ax-hvmulid 31358	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
19	5, 18	ax-mp 5	. . . . . . . 8 ⊢ (1 ·_ℎ 𝐴) = 𝐴
20	17, 19	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
21	20	adantr 485	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
22	8, 21	eqtr3d 2800	. . . . 5 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = 𝐴)
23	5, 1	hicli 31433	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
24		ax-hvmulass 31359	. . . . . . . . 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 1482	. . . . . . . 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 18	. . . . . . 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 11181	. . . . . . . . . 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 597	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
29	23, 9	divreczi 11948	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
30	28, 29	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)))
31	30	oveq1d 7425	. . . . . . 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 2800	. . . . . 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 485	. . . . 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 2800	. . . 4 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
35	34	ex 417	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
36	23, 9	divclzi 11945	. . . . 5 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ)
37	1	elexi 3477	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
38	37	snss 4750	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ ↔ {𝐵} ⊆ ℋ)
39	1, 38	mpbi 233	. . . . . . . . 9 ⊢ {𝐵} ⊆ ℋ
40		occl 31656	. . . . . . . . 9 ⊢ ({𝐵} ⊆ ℋ → (⊥‘{𝐵}) ∈ C_ℋ )
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ (⊥‘{𝐵}) ∈ C_ℋ
42	41	choccli 31659	. . . . . . 7 ⊢ (⊥‘(⊥‘{𝐵})) ∈ C_ℋ
43	42	chshii 31579	. . . . . 6 ⊢ (⊥‘(⊥‘{𝐵})) ∈ S_ℋ
44		h1did 31903	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (⊥‘(⊥‘{𝐵})))
45	1, 44	ax-mp 5	. . . . . 6 ⊢ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))
46		shmulcl 31570	. . . . . 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 1481	. . . . 5 ⊢ (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ → (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵})))
48	36, 47	syl 18	. . . 4 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵})))
49		eleq1 2851	. . . 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 250	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
51	35, 50	impbid 215	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
52	4, 51	sylbir 238	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 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ⊆ wss 3905 {csn 4589 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 0cc0 11095 1c1 11096 · cmul 11100 / cdiv 11866 ℋchba 31271 ·_ℎ csm 31273 ·_ih csp 31274 0_ℎc0v 31276 S_ℋ csh 31280 C_ℋ cch 31281 ⊥cort 31282

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175 ax-hilex 31351 ax-hfvadd 31352 ax-hvcom 31353 ax-hvass 31354 ax-hv0cl 31355 ax-hvaddid 31356 ax-hfvmul 31357 ax-hvmulid 31358 ax-hvmulass 31359 ax-hvdistr1 31360 ax-hvdistr2 31361 ax-hvmul0 31362 ax-hfi 31431 ax-his1 31434 ax-his2 31435 ax-his3 31436 ax-his4 31437 ax-hcompl 31554

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cn 23384 df-cnp 23385 df-lm 23386 df-haus 23472 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479 df-cau 25415 df-grpo 30845 df-gid 30846 df-ginv 30847 df-gdiv 30848 df-ablo 30897 df-vc 30911 df-nv 30944 df-va 30947 df-ba 30948 df-sm 30949 df-0v 30950 df-vs 30951 df-nmcv 30952 df-ims 30953 df-dip 31053 df-hnorm 31320 df-hvsub 31323 df-hlim 31324 df-hcau 31325 df-sh 31559 df-ch 31573 df-oc 31604

This theorem is referenced by: h1de2ctlem 31907 elspansn2 31919

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