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Mirrors > Home > HSE Home > Th. List > spansncol | Structured version Visualization version GIF version |
Description: The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spansncol | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·ℎ 𝐴)}) = (span‘{𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcl 10796 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) | |
2 | 1 | ancoms 462 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) |
3 | 2 | adantll 714 | . . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) |
4 | ax-hvmulass 29060 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) | |
5 | 4 | 3com13 1126 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
6 | 5 | 3expa 1120 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
7 | 6 | eqeq2d 2745 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴) ↔ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
8 | 7 | biimprd 251 | . . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → 𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴))) |
9 | oveq1 7209 | . . . . . . . 8 ⊢ (𝑧 = (𝑦 · 𝐵) → (𝑧 ·ℎ 𝐴) = ((𝑦 · 𝐵) ·ℎ 𝐴)) | |
10 | 9 | rspceeqv 3545 | . . . . . . 7 ⊢ (((𝑦 · 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴)) |
11 | 3, 8, 10 | syl6an 684 | . . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
12 | 11 | rexlimdva 3196 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
13 | 12 | 3adant3 1134 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
14 | divcl 11479 | . . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 / 𝐵) ∈ ℂ) | |
15 | 14 | 3expb 1122 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ) |
16 | 15 | adantlr 715 | . . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ) |
17 | simprl 771 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ) | |
18 | simplr 769 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℋ) | |
19 | ax-hvmulass 29060 | . . . . . . . . . . . . 13 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (((𝑧 / 𝐵) · 𝐵) ·ℎ 𝐴) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) | |
20 | 16, 17, 18, 19 | syl3anc 1373 | . . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·ℎ 𝐴) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) |
21 | divcan1 11482 | . . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) | |
22 | 21 | 3expb 1122 | . . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) |
23 | 22 | adantlr 715 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) |
24 | 23 | oveq1d 7217 | . . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·ℎ 𝐴) = (𝑧 ·ℎ 𝐴)) |
25 | 20, 24 | eqtr3d 2776 | . . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴)) = (𝑧 ·ℎ 𝐴)) |
26 | 25 | eqeq2d 2745 | . . . . . . . . . 10 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴)) ↔ 𝑥 = (𝑧 ·ℎ 𝐴))) |
27 | 26 | biimprd 251 | . . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·ℎ 𝐴) → 𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴)))) |
28 | oveq1 7209 | . . . . . . . . . 10 ⊢ (𝑦 = (𝑧 / 𝐵) → (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) | |
29 | 28 | rspceeqv 3545 | . . . . . . . . 9 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
30 | 16, 27, 29 | syl6an 684 | . . . . . . . 8 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
31 | 30 | exp43 440 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ → (𝐴 ∈ ℋ → (𝐵 ∈ ℂ → (𝐵 ≠ 0 → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))))))) |
32 | 31 | com4l 92 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐵 ∈ ℂ → (𝐵 ≠ 0 → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))))))) |
33 | 32 | 3imp 1113 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))))) |
34 | 33 | rexlimdv 3195 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
35 | 13, 34 | impbid 215 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
36 | hvmulcl 29066 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ℎ 𝐴) ∈ ℋ) | |
37 | 36 | ancoms 462 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·ℎ 𝐴) ∈ ℋ) |
38 | elspansn 29619 | . . . . 5 ⊢ ((𝐵 ·ℎ 𝐴) ∈ ℋ → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) | |
39 | 37, 38 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
40 | 39 | 3adant3 1134 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
41 | elspansn 29619 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) | |
42 | 41 | 3ad2ant1 1135 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
43 | 35, 40, 42 | 3bitr4d 314 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ 𝑥 ∈ (span‘{𝐴}))) |
44 | 43 | eqrdv 2732 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·ℎ 𝐴)}) = (span‘{𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∃wrex 3055 {csn 4531 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 0cc0 10712 · cmul 10717 / cdiv 11472 ℋchba 28972 ·ℎ csm 28974 spancspn 28985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cc 10032 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 ax-hilex 29052 ax-hfvadd 29053 ax-hvcom 29054 ax-hvass 29055 ax-hv0cl 29056 ax-hvaddid 29057 ax-hfvmul 29058 ax-hvmulid 29059 ax-hvmulass 29060 ax-hvdistr1 29061 ax-hvdistr2 29062 ax-hvmul0 29063 ax-hfi 29132 ax-his1 29135 ax-his2 29136 ax-his3 29137 ax-his4 29138 ax-hcompl 29255 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-rlim 15033 df-sum 15233 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-cn 22096 df-cnp 22097 df-lm 22098 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cfil 24124 df-cau 24125 df-cmet 24126 df-grpo 28546 df-gid 28547 df-ginv 28548 df-gdiv 28549 df-ablo 28598 df-vc 28612 df-nv 28645 df-va 28648 df-ba 28649 df-sm 28650 df-0v 28651 df-vs 28652 df-nmcv 28653 df-ims 28654 df-dip 28754 df-ssp 28775 df-ph 28866 df-cbn 28916 df-hnorm 29021 df-hba 29022 df-hvsub 29024 df-hlim 29025 df-hcau 29026 df-sh 29260 df-ch 29274 df-oc 29305 df-ch0 29306 df-span 29362 |
This theorem is referenced by: spansneleq 29623 superpos 30407 |
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