elspansn2 - Hilbert Space Explorer

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Theorem elspansn2 29502

Description: Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

**Assertion**
Ref	Expression
elspansn2	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))

**Proof of Theorem elspansn2**
Step	Hyp	Ref	Expression
1		spansn 29494	. . . 4 ⊢ (𝐵 ∈ ℋ → (span‘{𝐵}) = (⊥‘(⊥‘{𝐵})))
2	1	eleq2d 2818	. . 3 ⊢ (𝐵 ∈ ℋ → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
3	2	3ad2ant2 1135	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
4		eleq1 2820	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵}))))
5		id 22	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → 𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ))
6		oveq1 7177	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
7	6	oveq1d 7185	. . . . . . . 8 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)))
8	7	oveq1d 7185	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))
9	5, 8	eqeq12d 2754	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
10	4, 9	bibi12d 349	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))))
11	10	imbi2d 344	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → ((𝐵 ≠ 0_ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))) ↔ (𝐵 ≠ 0_ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))))
12		neeq1 2996	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ≠ 0_ℎ ↔ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ≠ 0_ℎ))
13		sneq 4526	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → {𝐵} = {if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})
14	13	fveq2d 6678	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘{𝐵}) = (⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)}))
15	14	fveq2d 6678	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘(⊥‘{𝐵})) = (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})))
16	15	eleq2d 2818	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)}))))
17		oveq2 7178	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
18		oveq1 7177	. . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ·_ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih 𝐵))
19		oveq2 7178	. . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
20	18, 19	eqtrd 2773	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ·_ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
21	17, 20	oveq12d 7188	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) = ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
22		id 22	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → 𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))
23	21, 22	oveq12d 7188	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))
24	23	eqeq2d 2749	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
25	16, 24	bibi12d 349	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)))))
26	12, 25	imbi12d 348	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → ((𝐵 ≠ 0_ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))) ↔ (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ≠ 0_ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))))
27		ifhvhv0 28957	. . . . 5 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
28		ifhvhv0 28957	. . . . 5 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
29	27, 28	h1de2bi 29489	. . . 4 ⊢ (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ≠ 0_ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))) ·_ℎ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))))
30	11, 26, 29	dedth2h 4473	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ≠ 0_ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))))
31	30	3impia 1118	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
32	3, 31	bitrd 282	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ifcif 4414 {csn 4516 ‘cfv 6339 (class class class)co 7170 / cdiv 11375 ℋchba 28854 ·_ℎ csm 28856 ·_ih csp 28857 0_ℎc0v 28859 ⊥cort 28865 spancspn 28867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cc 9935 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 ax-hilex 28934 ax-hfvadd 28935 ax-hvcom 28936 ax-hvass 28937 ax-hv0cl 28938 ax-hvaddid 28939 ax-hfvmul 28940 ax-hvmulid 28941 ax-hvmulass 28942 ax-hvdistr1 28943 ax-hvdistr2 28944 ax-hvmul0 28945 ax-hfi 29014 ax-his1 29017 ax-his2 29018 ax-his3 29019 ax-his4 29020 ax-hcompl 29137

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-oadd 8135 df-omul 8136 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-acn 9444 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-rlim 14936 df-sum 15136 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-mulg 18343 df-cntz 18565 df-cmn 19026 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-cn 21978 df-cnp 21979 df-lm 21980 df-haus 22066 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-xms 23073 df-ms 23074 df-tms 23075 df-cfil 24007 df-cau 24008 df-cmet 24009 df-grpo 28428 df-gid 28429 df-ginv 28430 df-gdiv 28431 df-ablo 28480 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-sm 28532 df-0v 28533 df-vs 28534 df-nmcv 28535 df-ims 28536 df-dip 28636 df-ssp 28657 df-ph 28748 df-cbn 28798 df-hnorm 28903 df-hba 28904 df-hvsub 28906 df-hlim 28907 df-hcau 28908 df-sh 29142 df-ch 29156 df-oc 29187 df-ch0 29188 df-span 29244

This theorem is referenced by: (None)

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