elspansn2 - Hilbert Space Explorer

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Theorem elspansn2 29908

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 29900	. . . 4 ⊢ (𝐵 ∈ ℋ → (span‘{𝐵}) = (⊥‘(⊥‘{𝐵})))
2	1	eleq2d 2825	. . 3 ⊢ (𝐵 ∈ ℋ → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
3	2	3ad2ant2 1132	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵}))))
4		eleq1 2827	. . . . . 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 7275	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
7	6	oveq1d 7283	. . . . . . . 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 7283	. . . . . . 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 2755	. . . . . 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 345	. . . . 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 340	. . . 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 3007	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ≠ 0_ℎ ↔ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ≠ 0_ℎ))
13		sneq 4576	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → {𝐵} = {if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})
14	13	fveq2d 6772	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘{𝐵}) = (⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)}))
15	14	fveq2d 6772	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘(⊥‘{𝐵})) = (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})))
16	15	eleq2d 2825	. . . . . 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 7276	. . . . . . . . 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 7275	. . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ·_ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih 𝐵))
19		oveq2 7276	. . . . . . . . . 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 2779	. . . . . . . . 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 7286	. . . . . . . 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 7286	. . . . . . 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 2750	. . . . . 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 345	. . . . 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 344	. . . 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 29363	. . . . 5 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
28		ifhvhv0 29363	. . . . 5 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
29	27, 28	h1de2bi 29895	. . . 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 4523	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ≠ 0_ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵))))
31	30	3impia 1115	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0_ℎ) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·_ih 𝐵) / (𝐵 ·_ih 𝐵)) ·_ℎ 𝐵)))
32	3, 31	bitrd 278	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 205 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ifcif 4464 {csn 4566 ‘cfv 6430 (class class class)co 7268 / cdiv 11615 ℋchba 29260 ·_ℎ csm 29262 ·_ih csp 29263 0_ℎc0v 29265 ⊥cort 29271 spancspn 29273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cc 10175 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 ax-hilex 29340 ax-hfvadd 29341 ax-hvcom 29342 ax-hvass 29343 ax-hv0cl 29344 ax-hvaddid 29345 ax-hfvmul 29346 ax-hvmulid 29347 ax-hvmulass 29348 ax-hvdistr1 29349 ax-hvdistr2 29350 ax-hvmul0 29351 ax-hfi 29420 ax-his1 29423 ax-his2 29424 ax-his3 29425 ax-his4 29426 ax-hcompl 29543

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-oadd 8285 df-omul 8286 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-acn 9684 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-fl 13493 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-rlim 15179 df-sum 15379 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-mulg 18682 df-cntz 18904 df-cmn 19369 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-fbas 20575 df-fg 20576 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-cls 22153 df-nei 22230 df-cn 22359 df-cnp 22360 df-lm 22361 df-haus 22447 df-tx 22694 df-hmeo 22887 df-fil 22978 df-fm 23070 df-flim 23071 df-flf 23072 df-xms 23454 df-ms 23455 df-tms 23456 df-cfil 24400 df-cau 24401 df-cmet 24402 df-grpo 28834 df-gid 28835 df-ginv 28836 df-gdiv 28837 df-ablo 28886 df-vc 28900 df-nv 28933 df-va 28936 df-ba 28937 df-sm 28938 df-0v 28939 df-vs 28940 df-nmcv 28941 df-ims 28942 df-dip 29042 df-ssp 29063 df-ph 29154 df-cbn 29204 df-hnorm 29309 df-hba 29310 df-hvsub 29312 df-hlim 29313 df-hcau 29314 df-sh 29548 df-ch 29562 df-oc 29593 df-ch0 29594 df-span 29650

This theorem is referenced by: (None)

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