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Mirrors > Home > MPE Home > Th. List > lsmssspx | Structured version Visualization version GIF version |
Description: Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.) |
Ref | Expression |
---|---|
lsmsp2.v | ⊢ 𝑉 = (Base‘𝑊) |
lsmsp2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsmsp2.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsmssspx.t | ⊢ (𝜑 → 𝑇 ⊆ 𝑉) |
lsmssspx.u | ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
lsmssspx.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
Ref | Expression |
---|---|
lsmssspx | ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmssspx.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lsmssspx.t | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑉) | |
3 | lsmsp2.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lsmsp2.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 3, 4 | lspssv 19477 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉) → (𝑁‘𝑇) ⊆ 𝑉) |
6 | 1, 2, 5 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑉) |
7 | lsmssspx.u | . . . 4 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) | |
8 | 3, 4 | lspssid 19479 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉) → 𝑇 ⊆ (𝑁‘𝑇)) |
9 | 1, 2, 8 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇)) |
10 | lsmsp2.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝑊) | |
11 | 3, 10 | lsmless1x 18530 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘𝑇) ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) ∧ 𝑇 ⊆ (𝑁‘𝑇)) → (𝑇 ⊕ 𝑈) ⊆ ((𝑁‘𝑇) ⊕ 𝑈)) |
12 | 1, 6, 7, 9, 11 | syl31anc 1353 | . . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ ((𝑁‘𝑇) ⊕ 𝑈)) |
13 | 3, 4 | lspssv 19477 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ⊆ 𝑉) |
14 | 1, 7, 13 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑈) ⊆ 𝑉) |
15 | 3, 4 | lspssid 19479 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ (𝑁‘𝑈)) |
16 | 1, 7, 15 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) |
17 | 3, 10 | lsmless2x 18531 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘𝑇) ⊆ 𝑉 ∧ (𝑁‘𝑈) ⊆ 𝑉) ∧ 𝑈 ⊆ (𝑁‘𝑈)) → ((𝑁‘𝑇) ⊕ 𝑈) ⊆ ((𝑁‘𝑇) ⊕ (𝑁‘𝑈))) |
18 | 1, 6, 14, 16, 17 | syl31anc 1353 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑇) ⊕ 𝑈) ⊆ ((𝑁‘𝑇) ⊕ (𝑁‘𝑈))) |
19 | 12, 18 | sstrd 3869 | . 2 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ ((𝑁‘𝑇) ⊕ (𝑁‘𝑈))) |
20 | 3, 4, 10 | lsmsp2 19581 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) → ((𝑁‘𝑇) ⊕ (𝑁‘𝑈)) = (𝑁‘(𝑇 ∪ 𝑈))) |
21 | 1, 2, 7, 20 | syl3anc 1351 | . 2 ⊢ (𝜑 → ((𝑁‘𝑇) ⊕ (𝑁‘𝑈)) = (𝑁‘(𝑇 ∪ 𝑈))) |
22 | 19, 21 | sseqtrd 3898 | 1 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∪ cun 3828 ⊆ wss 3830 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 LSSumclsm 18520 LModclmod 19356 LSpanclspn 19465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-cntz 18218 df-lsm 18522 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-lmod 19358 df-lss 19426 df-lsp 19466 |
This theorem is referenced by: djhsumss 37985 |
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