h1de2bi - Hilbert Space Explorer

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Theorem h1de2bi 30496

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 30041	. . . 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 2996	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 ↔ 𝐵 ≠ 0_ℎ)
5		h1de2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
6	5, 1	h1de2i 30495	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
7	6	adantl 482	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
8	7	oveq2d 7373	. . . . . 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 30023	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
10	9	recclzi 11880	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (1 / (𝐵 ·_ih 𝐵)) ∈ ℂ)
11		ax-hvmulass 29949	. . . . . . . . . . 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 1453	. . . . . . . . . 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 11109	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
15	14, 9	divcan1zi 11891	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) = 1)
16	15	oveq1d 7372	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = (1 ·_ℎ 𝐴))
17	13, 16	eqtr3d 2778	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = (1 ·_ℎ 𝐴))
18		ax-hvmulid 29948	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
19	5, 18	ax-mp 5	. . . . . . . 8 ⊢ (1 ·_ℎ 𝐴) = 𝐴
20	17, 19	eqtrdi 2792	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
21	20	adantr 481	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
22	8, 21	eqtr3d 2778	. . . . 5 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = 𝐴)
23	5, 1	hicli 30023	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
24		ax-hvmulass 29949	. . . . . . . . 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 1453	. . . . . . . 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 11137	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
29	23, 9	divreczi 11893	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
30	28, 29	eqtr4d 2779	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)))
31	30	oveq1d 7372	. . . . . . 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 2778	. . . . . 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 481	. . . . 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 2778	. . . 4 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
35	34	ex 413	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
36	23, 9	divclzi 11890	. . . . 5 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ)
37	1	elexi 3464	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
38	37	snss 4746	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ ↔ {𝐵} ⊆ ℋ)
39	1, 38	mpbi 229	. . . . . . . . 9 ⊢ {𝐵} ⊆ ℋ
40		occl 30246	. . . . . . . . 9 ⊢ ({𝐵} ⊆ ℋ → (⊥‘{𝐵}) ∈ C_ℋ )
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ (⊥‘{𝐵}) ∈ C_ℋ
42	41	choccli 30249	. . . . . . 7 ⊢ (⊥‘(⊥‘{𝐵})) ∈ C_ℋ
43	42	chshii 30169	. . . . . 6 ⊢ (⊥‘(⊥‘{𝐵})) ∈ S_ℋ
44		h1did 30493	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (⊥‘(⊥‘{𝐵})))
45	1, 44	ax-mp 5	. . . . . 6 ⊢ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))
46		shmulcl 30160	. . . . . 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 1452	. . . . 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 2825	. . . 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 246	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
51	35, 50	impbid 211	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
52	4, 51	sylbir 234	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3910 {csn 4586 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 · cmul 11056 / cdiv 11812 ℋchba 29861 ·_ℎ csm 29863 ·_ih csp 29864 0_ℎc0v 29866 S_ℋ csh 29870 C_ℋ cch 29871 ⊥cort 29872

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 ax-hcompl 30144

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cn 22578 df-cnp 22579 df-lm 22580 df-haus 22666 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-cau 24620 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-dip 29643 df-hnorm 29910 df-hvsub 29913 df-hlim 29914 df-hcau 29915 df-sh 30149 df-ch 30163 df-oc 30194

This theorem is referenced by: h1de2ctlem 30497 elspansn2 30509

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