elspansn2 - Hilbert Space Explorer

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Theorem elspansn2 32056

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 32048	. . . 4 ⊢ (𝐵 ∈ ℋ → (span‘{𝐵}) = (⊥‘(⊥‘{𝐵})))
2	1	eleq2d 2848	. . 3 ⊢ (𝐵 ∈ ℋ → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
3	2	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
4		eleq1 2850	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ (⊥‘(⊥‘{𝐵}))))
5		id 23	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → 𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ))
6		oveq1 7424	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
7	6	oveq1d 7432	. . . . . . . 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 7432	. . . . . . 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 2778	. . . . . 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 348	. . . . 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 343	. . . 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 3019	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ≠ 0_ℎ ↔ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ≠ 0_ℎ))
13		sneq 4597	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → {𝐵} = {if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})
14	13	fveq2d 6886	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘{𝐵}) = (⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)}))
15	14	fveq2d 6886	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘(⊥‘{𝐵})) = (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})))
16	15	eleq2d 2848	. . . . . 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 7425	. . . . . . . . 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 7424	. . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ·_ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih 𝐵))
19		oveq2 7425	. . . . . . . . . 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 2797	. . . . . . . . 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 7435	. . . . . . . 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 23	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → 𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ))
23	21, 22	oveq12d 7435	. . . . . . 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 2773	. . . . . 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 348	. . . . 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 347	. . . 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 31511	. . . . 5 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
28		ifhvhv0 31511	. . . . 5 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
29	27, 28	h1de2bi 32043	. . . 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 4545	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ≠ 0_ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))))
31	30	3impia 1135	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ifcif 4485 {csn 4587 ‘cfv 6537 (class class class)co 7417 / cdiv 11899 ℋchba 31408 ·_ℎ csm 31410 ·_ih csp 31411 0_ℎc0v 31413 ⊥cort 31419 spancspn 31421

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cc 10441 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207 ax-mulf 11208 ax-hilex 31488 ax-hfvadd 31489 ax-hvcom 31490 ax-hvass 31491 ax-hv0cl 31492 ax-hvaddid 31493 ax-hfvmul 31494 ax-hvmulid 31495 ax-hvmulass 31496 ax-hvdistr1 31497 ax-hvdistr2 31498 ax-hvmul0 31499 ax-hfi 31568 ax-his1 31571 ax-his2 31572 ax-his3 31573 ax-his4 31574 ax-hcompl 31691

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-acn 9951 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-rlim 15580 df-sum 15778 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-xrs 17594 df-qtop 17599 df-imas 17600 df-xps 17602 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-mulg 19197 df-cntz 19450 df-cmn 19915 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-fbas 21588 df-fg 21589 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-cn 23458 df-cnp 23459 df-lm 23460 df-haus 23546 df-tx 23794 df-hmeo 23987 df-fil 24078 df-fm 24170 df-flim 24171 df-flf 24172 df-xms 24552 df-ms 24553 df-tms 24554 df-cfil 25489 df-cau 25490 df-cmet 25491 df-grpo 30982 df-gid 30983 df-ginv 30984 df-gdiv 30985 df-ablo 31034 df-vc 31048 df-nv 31081 df-va 31084 df-ba 31085 df-sm 31086 df-0v 31087 df-vs 31088 df-nmcv 31089 df-ims 31090 df-dip 31190 df-ssp 31211 df-ph 31302 df-cbn 31352 df-hnorm 31457 df-hba 31458 df-hvsub 31460 df-hlim 31461 df-hcau 31462 df-sh 31696 df-ch 31710 df-oc 31741 df-ch0 31742 df-span 31798

This theorem is used by: (None)

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