![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > elspansn2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
elspansn2 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansn 31236 | . . . 4 ⊢ (𝐵 ∈ ℋ → (span‘{𝐵}) = (⊥‘(⊥‘{𝐵}))) | |
2 | 1 | eleq2d 2811 | . . 3 ⊢ (𝐵 ∈ ℋ → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵})))) |
3 | 2 | 3ad2ant2 1131 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0ℎ) → (𝐴 ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵})))) |
4 | eleq1 2813 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{𝐵})))) | |
5 | id 22 | . . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → 𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) | |
6 | oveq1 7408 | . . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵)) | |
7 | 6 | oveq1d 7416 | . . . . . . . 8 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) = ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵))) |
8 | 7 | oveq1d 7416 | . . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)) |
9 | 5, 8 | eqeq12d 2740 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) |
10 | 4, 9 | bibi12d 345 | . . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)))) |
11 | 10 | imbi2d 340 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐵 ≠ 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) ↔ (𝐵 ≠ 0ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))))) |
12 | neeq1 2995 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐵 ≠ 0ℎ ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ≠ 0ℎ)) | |
13 | sneq 4630 | . . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → {𝐵} = {if(𝐵 ∈ ℋ, 𝐵, 0ℎ)}) | |
14 | 13 | fveq2d 6885 | . . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (⊥‘{𝐵}) = (⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0ℎ)})) |
15 | 14 | fveq2d 6885 | . . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (⊥‘(⊥‘{𝐵})) = (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0ℎ)}))) |
16 | 15 | eleq2d 2811 | . . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0ℎ)})))) |
17 | oveq2 7409 | . . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) | |
18 | oveq1 7408 | . . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐵 ·ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih 𝐵)) | |
19 | oveq2 7409 | . . . . . . . . . 10 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) | |
20 | 18, 19 | eqtrd 2764 | . . . . . . . . 9 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐵 ·ih 𝐵) = (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) |
21 | 17, 20 | oveq12d 7419 | . . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) = ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))) |
22 | id 22 | . . . . . . . 8 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → 𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) | |
23 | 21, 22 | oveq12d 7419 | . . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ·ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) |
24 | 23 | eqeq2d 2735 | . . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ·ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))) |
25 | 16, 24 | bibi12d 345 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0ℎ)})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ·ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))))) |
26 | 12, 25 | imbi12d 344 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((𝐵 ≠ 0ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{𝐵})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ≠ 0ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0ℎ)})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ·ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))))) |
27 | ifhvhv0 30699 | . . . . 5 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
28 | ifhvhv0 30699 | . . . . 5 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ | |
29 | 27, 28 | h1de2bi 31231 | . . . 4 ⊢ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ≠ 0ℎ → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ (⊥‘(⊥‘{if(𝐵 ∈ ℋ, 𝐵, 0ℎ)})) ↔ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) = (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) / (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ·ih if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ·ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))) |
30 | 11, 26, 29 | dedth2h 4579 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ≠ 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)))) |
31 | 30 | 3impia 1114 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0ℎ) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) |
32 | 3, 31 | bitrd 279 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐵 ≠ 0ℎ) → (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 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ifcif 4520 {csn 4620 ‘cfv 6533 (class class class)co 7401 / cdiv 11867 ℋchba 30596 ·ℎ csm 30598 ·ih csp 30599 0ℎc0v 30601 ⊥cort 30607 spancspn 30609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 ax-hilex 30676 ax-hfvadd 30677 ax-hvcom 30678 ax-hvass 30679 ax-hv0cl 30680 ax-hvaddid 30681 ax-hfvmul 30682 ax-hvmulid 30683 ax-hvmulass 30684 ax-hvdistr1 30685 ax-hvdistr2 30686 ax-hvmul0 30687 ax-hfi 30756 ax-his1 30759 ax-his2 30760 ax-his3 30761 ax-his4 30762 ax-hcompl 30879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-acn 9932 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-cn 23041 df-cnp 23042 df-lm 23043 df-haus 23129 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cfil 25093 df-cau 25094 df-cmet 25095 df-grpo 30170 df-gid 30171 df-ginv 30172 df-gdiv 30173 df-ablo 30222 df-vc 30236 df-nv 30269 df-va 30272 df-ba 30273 df-sm 30274 df-0v 30275 df-vs 30276 df-nmcv 30277 df-ims 30278 df-dip 30378 df-ssp 30399 df-ph 30490 df-cbn 30540 df-hnorm 30645 df-hba 30646 df-hvsub 30648 df-hlim 30649 df-hcau 30650 df-sh 30884 df-ch 30898 df-oc 30929 df-ch0 30930 df-span 30986 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |