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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 11135 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) | |
2 | 1 | ancoms 459 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) |
3 | 2 | adantll 712 | . . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑦 · 𝐵) ∈ ℂ) |
4 | ax-hvmulass 29949 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) | |
5 | 4 | 3com13 1124 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑦 · 𝐵) ·ℎ 𝐴) = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
6 | 5 | 3expa 1118 | . . . . . . . . 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 7364 | . . . . . . . 8 ⊢ (𝑧 = (𝑦 · 𝐵) → (𝑧 ·ℎ 𝐴) = ((𝑦 · 𝐵) ·ℎ 𝐴)) | |
10 | 9 | rspceeqv 3595 | . . . . . . 7 ⊢ (((𝑦 · 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑦 · 𝐵) ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴)) |
11 | 3, 8, 10 | syl6an 682 | . . . . . 6 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℂ) → (𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
12 | 11 | rexlimdva 3152 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
13 | 12 | 3adant3 1132 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) → ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) |
14 | divcl 11819 | . . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 / 𝐵) ∈ ℂ) | |
15 | 14 | 3expb 1120 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ) |
16 | 15 | adantlr 713 | . . . . . . . . 9 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝑧 / 𝐵) ∈ ℂ) |
17 | simprl 769 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℂ) | |
18 | simplr 767 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℋ) | |
19 | ax-hvmulass 29949 | . . . . . . . . . . . . 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 1371 | . . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝑧 / 𝐵) · 𝐵) ·ℎ 𝐴) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) |
21 | divcan1 11822 | . . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) | |
22 | 21 | 3expb 1120 | . . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) |
23 | 22 | adantlr 713 | . . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝑧 / 𝐵) · 𝐵) = 𝑧) |
24 | 23 | oveq1d 7372 | . . . . . . . . . . . 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 7364 | . . . . . . . . . 10 ⊢ (𝑦 = (𝑧 / 𝐵) → (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)) = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) | |
29 | 28 | rspceeqv 3595 | . . . . . . . . 9 ⊢ (((𝑧 / 𝐵) ∈ ℂ ∧ 𝑥 = ((𝑧 / 𝐵) ·ℎ (𝐵 ·ℎ 𝐴))) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))) |
30 | 16, 27, 29 | syl6an 682 | . . . . . . . 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 1111 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·ℎ 𝐴) → ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴))))) |
34 | 33 | rexlimdv 3150 | . . . 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 29955 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ℎ 𝐴) ∈ ℋ) | |
37 | 36 | ancoms 459 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) → (𝐵 ·ℎ 𝐴) ∈ ℋ) |
38 | elspansn 30508 | . . . . 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 1132 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝑥 ∈ (span‘{(𝐵 ·ℎ 𝐴)}) ↔ ∃𝑦 ∈ ℂ 𝑥 = (𝑦 ·ℎ (𝐵 ·ℎ 𝐴)))) |
41 | elspansn 30508 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴))) | |
42 | 41 | 3ad2ant1 1133 | . . 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 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 {csn 4586 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 · cmul 11056 / cdiv 11812 ℋchba 29861 ·ℎ csm 29863 spancspn 29874 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 ax-hcompl 30144 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-cn 22578 df-cnp 22579 df-lm 22580 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-dip 29643 df-ssp 29664 df-ph 29755 df-cbn 29805 df-hnorm 29910 df-hba 29911 df-hvsub 29913 df-hlim 29914 df-hcau 29915 df-sh 30149 df-ch 30163 df-oc 30194 df-ch0 30195 df-span 30251 |
This theorem is referenced by: spansneleq 30512 superpos 31296 |
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