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Mirrors > Home > MPE Home > Th. List > lsppr | Structured version Visualization version GIF version |
Description: Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.) |
Ref | Expression |
---|---|
lsppr.v | ⊢ 𝑉 = (Base‘𝑊) |
lsppr.a | ⊢ + = (+g‘𝑊) |
lsppr.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lsppr.k | ⊢ 𝐾 = (Base‘𝐹) |
lsppr.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lsppr.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsppr.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsppr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsppr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lsppr | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4564 | . . 3 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
2 | 1 | fveq2i 6668 | . 2 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
3 | lsppr.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lsppr.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
5 | 4 | snssd 4736 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
6 | lsppr.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | 6 | snssd 4736 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
8 | lsppr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | lsppr.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 8, 9 | lspun 19753 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
11 | 3, 5, 7, 10 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
12 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | 8, 12, 9 | lspsncl 19743 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
14 | 3, 4, 13 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 8, 12, 9 | lspsncl 19743 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 3, 6, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | eqid 2821 | . . . . 5 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
18 | 12, 9, 17 | lsmsp 19852 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
19 | 3, 14, 16, 18 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
20 | lsppr.a | . . . . 5 ⊢ + = (+g‘𝑊) | |
21 | lsppr.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
22 | lsppr.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
23 | lsppr.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
24 | 8, 20, 21, 22, 23, 17, 9, 3, 4, 6 | lsmspsn 19850 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
25 | 24 | abbi2dv 2950 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
26 | 11, 19, 25 | 3eqtr2d 2862 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
27 | 2, 26 | syl5eq 2868 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ∪ cun 3934 ⊆ wss 3936 {csn 4561 {cpr 4563 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 LSSumclsm 18753 LModclmod 19628 LSubSpclss 19697 LSpanclspn 19737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 df-lss 19698 df-lsp 19738 |
This theorem is referenced by: lspprel 19860 |
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