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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lspindp5 | Structured version Visualization version GIF version |
Description: Obtain an independent vector set 𝑈, 𝑋, 𝑌 from a vector 𝑈 dependent on 𝑋 and 𝑍 and another independent set 𝑍, 𝑋, 𝑌. (Here we don't show the (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) part of the independence, which passes straight through. We also don't show nonzero vector requirements that are redundant for this theorem. Different orderings can be obtained using lspexch 19894 and prcom 4628.) (Contributed by NM, 4-May-2015.) |
Ref | Expression |
---|---|
lspindp5.v | ⊢ 𝑉 = (Base‘𝑊) |
lspindp5.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspindp5.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspindp5.y | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspindp5.x | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspindp5.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
lspindp5.e | ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑋, 𝑈})) |
lspindp5.m | ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
lspindp5 | ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑁‘{𝑋, 𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspindp5.m | . . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) | |
2 | lspindp5.e | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑋, 𝑈})) | |
3 | ssel 3908 | . . . 4 ⊢ ((𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) → (𝑍 ∈ (𝑁‘{𝑋, 𝑈}) → 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))) | |
4 | 2, 3 | syl5com 31 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) → 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))) |
5 | 1, 4 | mtod 201 | . 2 ⊢ (𝜑 → ¬ (𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌})) |
6 | lspindp5.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lveclmod 19871 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
9 | lspindp5.y | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
10 | lspindp5.x | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | prssi 4714 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
12 | 9, 10, 11 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉) |
13 | snsspr1 4707 | . . . . . . 7 ⊢ {𝑋} ⊆ {𝑋, 𝑌} | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ {𝑋, 𝑌}) |
15 | lspindp5.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
16 | lspindp5.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊) | |
17 | 15, 16 | lspss 19749 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
18 | 8, 12, 14, 17 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
19 | 18 | biantrurd 536 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) ↔ ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}) ∧ (𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌})))) |
20 | eqid 2798 | . . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
21 | 20 | lsssssubg 19723 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
22 | 8, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
23 | 15, 20, 16 | lspsncl 19742 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
24 | 8, 9, 23 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
25 | 22, 24 | sseldd 3916 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
26 | lspindp5.u | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
27 | 15, 20, 16 | lspsncl 19742 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑉) → (𝑁‘{𝑈}) ∈ (LSubSp‘𝑊)) |
28 | 8, 26, 27 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑈}) ∈ (LSubSp‘𝑊)) |
29 | 22, 28 | sseldd 3916 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑈}) ∈ (SubGrp‘𝑊)) |
30 | 15, 20, 16, 8, 9, 10 | lspprcl 19743 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊)) |
31 | 22, 30 | sseldd 3916 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑊)) |
32 | eqid 2798 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
33 | 32 | lsmlub 18782 | . . . . 5 ⊢ (((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑈}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑊)) → (((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}) ∧ (𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌})) ↔ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈})) ⊆ (𝑁‘{𝑋, 𝑌}))) |
34 | 25, 29, 31, 33 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}) ∧ (𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌})) ↔ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈})) ⊆ (𝑁‘{𝑋, 𝑌}))) |
35 | 19, 34 | bitrd 282 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) ↔ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈})) ⊆ (𝑁‘{𝑋, 𝑌}))) |
36 | 15, 20, 16, 8, 30, 26 | lspsnel5 19760 | . . 3 ⊢ (𝜑 → (𝑈 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}))) |
37 | 15, 16, 32, 8, 9, 26 | lsmpr 19854 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑈}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈}))) |
38 | 37 | sseq1d 3946 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) ↔ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈})) ⊆ (𝑁‘{𝑋, 𝑌}))) |
39 | 35, 36, 38 | 3bitr4d 314 | . 2 ⊢ (𝜑 → (𝑈 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}))) |
40 | 5, 39 | mtbird 328 | 1 ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑁‘{𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 {csn 4525 {cpr 4527 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 SubGrpcsubg 18265 LSSumclsm 18751 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LVecclvec 19867 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 |
This theorem is referenced by: mapdh8b 39076 |
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