h1de2bi - Hilbert Space Explorer

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Theorem h1de2bi 27797

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 27340	. . . 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 2834	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 ↔ 𝐵 ≠ 0_ℎ)
5		h1de2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ ℋ
6	5, 1	h1de2i 27796	. . . . . . . 8 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
7	6	adantl 481	. . . . . . 7 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴) = ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵))
8	7	oveq2d 6565	. . . . . 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 27322	. . . . . . . . . . 11 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
10	9	recclzi 10629	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (1 / (𝐵 ·_ih 𝐵)) ∈ ℂ)
11		ax-hvmulass 27248	. . . . . . . . . . 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 1408	. . . . . . . . . 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 9873	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
15	14, 9	divcan1zi 10640	. . . . . . . . . 10 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) = 1)
16	15	oveq1d 6564	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (((1 / (𝐵 ·_ih 𝐵)) · (𝐵 ·_ih 𝐵)) ·_ℎ 𝐴) = (1 ·_ℎ 𝐴))
17	13, 16	eqtr3d 2646	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = (1 ·_ℎ 𝐴))
18		ax-hvmulid 27247	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
19	5, 18	ax-mp 5	. . . . . . . 8 ⊢ (1 ·_ℎ 𝐴) = 𝐴
20	17, 19	syl6eq 2660	. . . . . . 7 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
21	20	adantr 480	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐵 ·_ih 𝐵) ·_ℎ 𝐴)) = 𝐴)
22	8, 21	eqtr3d 2646	. . . . 5 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → ((1 / (𝐵 ·_ih 𝐵)) ·_ℎ ((𝐴 ·_ih 𝐵) ·_ℎ 𝐵)) = 𝐴)
23	5, 1	hicli 27322	. . . . . . . . 9 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
24		ax-hvmulass 27248	. . . . . . . . 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 1408	. . . . . . . 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 9901	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
29	23, 9	divreczi 10642	. . . . . . . . 9 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) · (1 / (𝐵 ·_ih 𝐵))))
30	28, 29	eqtr4d 2647	. . . . . . . 8 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((1 / (𝐵 ·_ih 𝐵)) · (𝐴 ·_ih 𝐵)) = ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)))
31	30	oveq1d 6564	. . . . . . 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 2646	. . . . . 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 480	. . . . 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 2646	. . . 4 ⊢ (((𝐵 ·_ih 𝐵) ≠ 0 ∧ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
35	34	ex 449	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
36	23, 9	divclzi 10639	. . . . 5 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ∈ ℂ)
37	1	elexi 3186	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
38	37	snss 4259	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ ↔ {𝐵} ⊆ ℋ)
39	1, 38	mpbi 219	. . . . . . . . 9 ⊢ {𝐵} ⊆ ℋ
40		occl 27547	. . . . . . . . 9 ⊢ ({𝐵} ⊆ ℋ → (⊥‘{𝐵}) ∈ C_ℋ )
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ (⊥‘{𝐵}) ∈ C_ℋ
42	41	choccli 27550	. . . . . . 7 ⊢ (⊥‘(⊥‘{𝐵})) ∈ C_ℋ
43	42	chshii 27468	. . . . . 6 ⊢ (⊥‘(⊥‘{𝐵})) ∈ S_ℋ
44		h1did 27794	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (⊥‘(⊥‘{𝐵})))
45	1, 44	ax-mp 5	. . . . . 6 ⊢ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))
46		shmulcl 27459	. . . . . 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 1407	. . . . 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 2676	. . . 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 236	. . 3 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
51	35, 50	impbid 201	. 2 ⊢ ((𝐵 ·_ih 𝐵) ≠ 0 → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
52	4, 51	sylbir 224	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 {csn 4125 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 ℋchil 27160 ·_ℎ csm 27162 ·_ih csp 27163 0_ℎc0v 27165 S_ℋ csh 27169 C_ℋ cch 27170 ⊥cort 27171

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 ax-hcompl 27443

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-lm 20843 df-haus 20929 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cau 22862 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-dip 26940 df-hnorm 27209 df-hvsub 27212 df-hlim 27213 df-hcau 27214 df-sh 27448 df-ch 27462 df-oc 27493

This theorem is referenced by: h1de2ctlem 27798 elspansn2 27810

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