elspansn2 - Hilbert Space Explorer

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Theorem elspansn2 31658

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 31650	. . . 4 ⊢ (𝐵 ∈ ℋ → (span‘{𝐵}) = (⊥‘(⊥‘{𝐵})))
2	1	eleq2d 2823	. . 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 2825	. . . . . 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 7365	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) → (𝐴 ·_ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ·_ih 𝐵))
7	6	oveq1d 7373	. . . . . . . 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 7373	. . . . . . 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 2753	. . . . . 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 2995	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ≠ 0_ℎ ↔ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ≠ 0_ℎ))
13		sneq 4578	. . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → {𝐵} = {if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})
14	13	fveq2d 6836	. . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘{𝐵}) = (⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)}))
15	14	fveq2d 6836	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (⊥‘(⊥‘{𝐵})) = (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0_ℎ)})))
16	15	eleq2d 2823	. . . . . 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 7366	. . . . . . . . 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 7365	. . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) → (𝐵 ·_ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ·_ih 𝐵))
19		oveq2 7366	. . . . . . . . . 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 2772	. . . . . . . . 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 7376	. . . . . . . 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 7376	. . . . . . 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 2748	. . . . . 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 31113	. . . . 5 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0_ℎ) ∈ ℋ
28		ifhvhv0 31113	. . . . 5 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0_ℎ) ∈ ℋ
29	27, 28	h1de2bi 31645	. . . 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 4527	. . 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 279	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 206 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ifcif 4467 {csn 4568 ‘cfv 6490 (class class class)co 7358 / cdiv 11796 ℋchba 31010 ·_ℎ csm 31012 ·_ih csp 31013 0_ℎc0v 31015 ⊥cort 31021 spancspn 31023

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cc 10346 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107 ax-hilex 31090 ax-hfvadd 31091 ax-hvcom 31092 ax-hvass 31093 ax-hv0cl 31094 ax-hvaddid 31095 ax-hfvmul 31096 ax-hvmulid 31097 ax-hvmulass 31098 ax-hvdistr1 31099 ax-hvdistr2 31100 ax-hvmul0 31101 ax-hfi 31170 ax-his1 31173 ax-his2 31174 ax-his3 31175 ax-his4 31176 ax-hcompl 31293

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-seq 13953 df-exp 14013 df-hash 14282 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15439 df-rlim 15440 df-sum 15638 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-cn 23201 df-cnp 23202 df-lm 23203 df-haus 23289 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cfil 25231 df-cau 25232 df-cmet 25233 df-grpo 30584 df-gid 30585 df-ginv 30586 df-gdiv 30587 df-ablo 30636 df-vc 30650 df-nv 30683 df-va 30686 df-ba 30687 df-sm 30688 df-0v 30689 df-vs 30690 df-nmcv 30691 df-ims 30692 df-dip 30792 df-ssp 30813 df-ph 30904 df-cbn 30954 df-hnorm 31059 df-hba 31060 df-hvsub 31062 df-hlim 31063 df-hcau 31064 df-sh 31298 df-ch 31312 df-oc 31343 df-ch0 31344 df-span 31400

This theorem is referenced by: (None)

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