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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 11034 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) | |
2 | 1 | ancoms 459 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) |
3 | 2 | adantll 711 | . . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) |
4 | ax-hvmulass 29501 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) | |
5 | 4 | 3com13 1123 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
6 | 5 | 3expa 1117 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
7 | 6 | eqeq2d 2747 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴) ↔ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
8 | 7 | biimprd 247 | . . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → 𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴))) |
9 | oveq1 7323 | . . . . . . . 8 ⊢ (𝑧 = (𝑦 · 𝐵) → (𝑧 ·ℎ 𝐴) = ((𝑦 · 𝐵) ·ℎ 𝐴)) | |
10 | 9 | rspceeqv 3583 | . . . . . . 7 ⊢ (((𝑦 · 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴)) |
11 | 3, 8, 10 | syl6an 681 | . . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
12 | 11 | rexlimdva 3148 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
13 | 12 | 3adant3 1131 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
14 | divcl 11718 | . . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 / 𝐵) ∈ ℂ) | |
15 | 14 | 3expb 1119 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ) |
16 | 15 | adantlr 712 | . . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ) |
17 | simprl 768 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ) | |
18 | simplr 766 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℋ) | |
19 | ax-hvmulass 29501 | . . . . . . . . . . . . 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 1370 | . . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·ℎ 𝐴) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) |
21 | divcan1 11721 | . . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) | |
22 | 21 | 3expb 1119 | . . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) |
23 | 22 | adantlr 712 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) |
24 | 23 | oveq1d 7331 | . . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·ℎ 𝐴) = (𝑧 ·ℎ 𝐴)) |
25 | 20, 24 | eqtr3d 2778 | . . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴)) = (𝑧 ·ℎ 𝐴)) |
26 | 25 | eqeq2d 2747 | . . . . . . . . . 10 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴)) ↔ 𝑥 = (𝑧 ·ℎ 𝐴))) |
27 | 26 | biimprd 247 | . . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·ℎ 𝐴) → 𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴)))) |
28 | oveq1 7323 | . . . . . . . . . 10 ⊢ (𝑦 = (𝑧 / 𝐵) → (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) | |
29 | 28 | rspceeqv 3583 | . . . . . . . . 9 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
30 | 16, 27, 29 | syl6an 681 | . . . . . . . 8 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
31 | 30 | exp43 437 | . . . . . . 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 1110 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))))) |
34 | 33 | rexlimdv 3146 | . . . 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 211 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
36 | hvmulcl 29507 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ℎ 𝐴) ∈ ℋ) | |
37 | 36 | ancoms 459 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·ℎ 𝐴) ∈ ℋ) |
38 | elspansn 30060 | . . . . 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 1131 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
41 | elspansn 30060 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) | |
42 | 41 | 3ad2ant1 1132 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
43 | 35, 40, 42 | 3bitr4d 310 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ 𝑥 ∈ (span‘{𝐴}))) |
44 | 43 | eqrdv 2734 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (span‘{(𝐵 ·ℎ 𝐴)}) = (span‘{𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∃wrex 3070 {csn 4570 ‘cfv 6465 (class class class)co 7316 ℂcc 10948 0cc0 10950 · cmul 10955 / cdiv 11711 ℋchba 29413 ·ℎ csm 29415 spancspn 29426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-inf2 9476 ax-cc 10270 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028 ax-addf 11029 ax-mulf 11030 ax-hilex 29493 ax-hfvadd 29494 ax-hvcom 29495 ax-hvass 29496 ax-hv0cl 29497 ax-hvaddid 29498 ax-hfvmul 29499 ax-hvmulid 29500 ax-hvmulass 29501 ax-hvdistr1 29502 ax-hvdistr2 29503 ax-hvmul0 29504 ax-hfi 29573 ax-his1 29576 ax-his2 29577 ax-his3 29578 ax-his4 29579 ax-hcompl 29696 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-of 7574 df-om 7759 df-1st 7877 df-2nd 7878 df-supp 8026 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-2o 8346 df-oadd 8349 df-omul 8350 df-er 8547 df-map 8666 df-pm 8667 df-ixp 8735 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-fsupp 9205 df-fi 9246 df-sup 9277 df-inf 9278 df-oi 9345 df-card 9774 df-acn 9777 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-z 12399 df-dec 12517 df-uz 12662 df-q 12768 df-rp 12810 df-xneg 12927 df-xadd 12928 df-xmul 12929 df-ioo 13162 df-ico 13164 df-icc 13165 df-fz 13319 df-fzo 13462 df-fl 13591 df-seq 13801 df-exp 13862 df-hash 14124 df-cj 14886 df-re 14887 df-im 14888 df-sqrt 15022 df-abs 15023 df-clim 15273 df-rlim 15274 df-sum 15474 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-ress 17016 df-plusg 17049 df-mulr 17050 df-starv 17051 df-sca 17052 df-vsca 17053 df-ip 17054 df-tset 17055 df-ple 17056 df-ds 17058 df-unif 17059 df-hom 17060 df-cco 17061 df-rest 17207 df-topn 17208 df-0g 17226 df-gsum 17227 df-topgen 17228 df-pt 17229 df-prds 17232 df-xrs 17287 df-qtop 17292 df-imas 17293 df-xps 17295 df-mre 17369 df-mrc 17370 df-acs 17372 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-submnd 18505 df-mulg 18774 df-cntz 18996 df-cmn 19460 df-psmet 20669 df-xmet 20670 df-met 20671 df-bl 20672 df-mopn 20673 df-fbas 20674 df-fg 20675 df-cnfld 20678 df-top 22123 df-topon 22140 df-topsp 22162 df-bases 22176 df-cld 22250 df-ntr 22251 df-cls 22252 df-nei 22329 df-cn 22458 df-cnp 22459 df-lm 22460 df-haus 22546 df-tx 22793 df-hmeo 22986 df-fil 23077 df-fm 23169 df-flim 23170 df-flf 23171 df-xms 23553 df-ms 23554 df-tms 23555 df-cfil 24499 df-cau 24500 df-cmet 24501 df-grpo 28987 df-gid 28988 df-ginv 28989 df-gdiv 28990 df-ablo 29039 df-vc 29053 df-nv 29086 df-va 29089 df-ba 29090 df-sm 29091 df-0v 29092 df-vs 29093 df-nmcv 29094 df-ims 29095 df-dip 29195 df-ssp 29216 df-ph 29307 df-cbn 29357 df-hnorm 29462 df-hba 29463 df-hvsub 29465 df-hlim 29466 df-hcau 29467 df-sh 29701 df-ch 29715 df-oc 29746 df-ch0 29747 df-span 29803 |
This theorem is referenced by: spansneleq 30064 superpos 30848 |
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