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Mirrors > Home > MPE Home > Th. List > lsmsp | Structured version Visualization version GIF version |
Description: Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
Ref | Expression |
---|---|
lsmsp.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsmsp.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsmsp.p | ⊢ ⊕ = (LSSum‘𝑊) |
Ref | Expression |
---|---|
lsmsp | ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) = (𝑁‘(𝑇 ∪ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1138 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ LMod) | |
2 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | lsmsp.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | 2, 3 | lssss 19927 | . . . . . . 7 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ⊆ (Base‘𝑊)) |
5 | 4 | 3ad2ant2 1136 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑇 ⊆ (Base‘𝑊)) |
6 | 2, 3 | lssss 19927 | . . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
7 | 6 | 3ad2ant3 1137 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (Base‘𝑊)) |
8 | 5, 7 | unssd 4086 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∪ 𝑈) ⊆ (Base‘𝑊)) |
9 | lsmsp.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 2, 9 | lspssid 19976 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ∪ 𝑈) ⊆ (Base‘𝑊)) → (𝑇 ∪ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
11 | 1, 8, 10 | syl2anc 587 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∪ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
12 | 11 | unssad 4087 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑇 ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
13 | 11 | unssbd 4088 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
14 | 3 | lsssssubg 19949 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
15 | 14 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑆 ⊆ (SubGrp‘𝑊)) |
16 | simp2 1139 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑇 ∈ 𝑆) | |
17 | 15, 16 | sseldd 3888 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑇 ∈ (SubGrp‘𝑊)) |
18 | simp3 1140 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ 𝑆) | |
19 | 15, 18 | sseldd 3888 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
20 | 2, 3, 9 | lspcl 19967 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ∪ 𝑈) ⊆ (Base‘𝑊)) → (𝑁‘(𝑇 ∪ 𝑈)) ∈ 𝑆) |
21 | 1, 8, 20 | syl2anc 587 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑁‘(𝑇 ∪ 𝑈)) ∈ 𝑆) |
22 | 15, 21 | sseldd 3888 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑁‘(𝑇 ∪ 𝑈)) ∈ (SubGrp‘𝑊)) |
23 | lsmsp.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝑊) | |
24 | 23 | lsmlub 19008 | . . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑁‘(𝑇 ∪ 𝑈)) ∈ (SubGrp‘𝑊)) → ((𝑇 ⊆ (𝑁‘(𝑇 ∪ 𝑈)) ∧ 𝑈 ⊆ (𝑁‘(𝑇 ∪ 𝑈))) ↔ (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈)))) |
25 | 17, 19, 22, 24 | syl3anc 1373 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → ((𝑇 ⊆ (𝑁‘(𝑇 ∪ 𝑈)) ∧ 𝑈 ⊆ (𝑁‘(𝑇 ∪ 𝑈))) ↔ (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈)))) |
26 | 12, 13, 25 | mpbi2and 712 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
27 | 3, 23 | lsmcl 20074 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆) |
28 | 23 | lsmunss 19002 | . . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑇 ∪ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) |
29 | 17, 19, 28 | syl2anc 587 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∪ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) |
30 | 3, 9 | lspssp 19979 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑇 ⊕ 𝑈) ∈ 𝑆 ∧ (𝑇 ∪ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) → (𝑁‘(𝑇 ∪ 𝑈)) ⊆ (𝑇 ⊕ 𝑈)) |
31 | 1, 27, 29, 30 | syl3anc 1373 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑁‘(𝑇 ∪ 𝑈)) ⊆ (𝑇 ⊕ 𝑈)) |
32 | 26, 31 | eqssd 3904 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) = (𝑁‘(𝑇 ∪ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∪ cun 3851 ⊆ wss 3853 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 SubGrpcsubg 18491 LSSumclsm 18977 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-ress 16674 df-plusg 16762 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-cntz 18665 df-lsm 18979 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-lmod 19855 df-lss 19923 df-lsp 19963 |
This theorem is referenced by: lsmsp2 20078 lsmpr 20080 lsppr 20084 lsmidllsp 31256 islshpsm 36680 lshpnel2N 36685 lkrlsp3 36804 djhlsmcl 39114 dochsatshp 39151 |
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