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Mirrors > Home > MPE Home > Th. List > lspvadd | Structured version Visualization version GIF version |
Description: The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
Ref | Expression |
---|---|
lspvadd.v | ⊢ 𝑉 = (Base‘𝑊) |
lspvadd.a | ⊢ + = (+g‘𝑊) |
lspvadd.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspvadd | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . 2 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | lspvadd.n | . 2 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | simp1 1138 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑊 ∈ LMod) | |
4 | prssi 4720 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
5 | 4 | 3adant1 1132 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) |
6 | lspvadd.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
7 | 6, 1, 2 | lspcl 19967 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊)) |
8 | 3, 5, 7 | syl2anc 587 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊)) |
9 | 6, 2 | lspssid 19976 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → {𝑋, 𝑌} ⊆ (𝑁‘{𝑋, 𝑌})) |
10 | 3, 5, 9 | syl2anc 587 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ (𝑁‘{𝑋, 𝑌})) |
11 | prssg 4718 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ (𝑁‘{𝑋, 𝑌}) ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑌})) ↔ {𝑋, 𝑌} ⊆ (𝑁‘{𝑋, 𝑌}))) | |
12 | 11 | 3adant1 1132 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ (𝑁‘{𝑋, 𝑌}) ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑌})) ↔ {𝑋, 𝑌} ⊆ (𝑁‘{𝑋, 𝑌}))) |
13 | 10, 12 | mpbird 260 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ (𝑁‘{𝑋, 𝑌}) ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑌}))) |
14 | lspvadd.a | . . . 4 ⊢ + = (+g‘𝑊) | |
15 | 14, 1 | lssvacl 19945 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊)) ∧ (𝑋 ∈ (𝑁‘{𝑋, 𝑌}) ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑌}))) → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
16 | 3, 8, 13, 15 | syl21anc 838 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
17 | 1, 2, 3, 8, 16 | lspsnel5a 19987 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 {csn 4527 {cpr 4529 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 LModclmod 19853 LSubSpclss 19922 LSpanclspn 19962 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mgp 19459 df-ur 19471 df-ring 19518 df-lmod 19855 df-lss 19923 df-lsp 19963 |
This theorem is referenced by: lspsntri 20088 lsmsat 36708 mapdindp3 39422 |
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