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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 21010 and prcom 4732.) (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 3971 | . . . 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 197 | . 2 ⊢ (𝜑 → ¬ (𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌})) |
6 | lspindp5.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lveclmod 20984 | . . . . . . 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 4820 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
12 | 9, 10, 11 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉) |
13 | snsspr1 4813 | . . . . . . 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 20861 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
18 | 8, 12, 14, 17 | syl3anc 1369 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
19 | 18 | biantrurd 532 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) ↔ ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}) ∧ (𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌})))) |
20 | eqid 2727 | . . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
21 | 20 | lsssssubg 20835 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
22 | 8, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
23 | 15, 20, 16 | lspsncl 20854 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
24 | 8, 9, 23 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
25 | 22, 24 | sseldd 3979 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
26 | lspindp5.u | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
27 | 15, 20, 16 | lspsncl 20854 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑉) → (𝑁‘{𝑈}) ∈ (LSubSp‘𝑊)) |
28 | 8, 26, 27 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑈}) ∈ (LSubSp‘𝑊)) |
29 | 22, 28 | sseldd 3979 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑈}) ∈ (SubGrp‘𝑊)) |
30 | 15, 20, 16, 8, 9, 10 | lspprcl 20855 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊)) |
31 | 22, 30 | sseldd 3979 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑊)) |
32 | eqid 2727 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
33 | 32 | lsmlub 19612 | . . . . 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 1369 | . . . 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 279 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) ↔ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈})) ⊆ (𝑁‘{𝑋, 𝑌}))) |
36 | 15, 20, 16, 8, 30, 26 | lspsnel5 20872 | . . 3 ⊢ (𝜑 → (𝑈 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}))) |
37 | 15, 16, 32, 8, 9, 26 | lsmpr 20967 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑈}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈}))) |
38 | 37 | sseq1d 4009 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}) ↔ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑈})) ⊆ (𝑁‘{𝑋, 𝑌}))) |
39 | 35, 36, 38 | 3bitr4d 311 | . 2 ⊢ (𝜑 → (𝑈 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑋, 𝑈}) ⊆ (𝑁‘{𝑋, 𝑌}))) |
40 | 5, 39 | mtbird 325 | 1 ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑁‘{𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 {csn 4624 {cpr 4626 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 SubGrpcsubg 19068 LSSumclsm 19582 LModclmod 20736 LSubSpclss 20808 LSpanclspn 20848 LVecclvec 20980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-ur 20115 df-ring 20168 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 |
This theorem is referenced by: mapdh8b 41242 |
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