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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp0 | Structured version Visualization version GIF version |
Description: Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
Ref | Expression |
---|---|
mapdindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
mapdindp1.p | ⊢ + = (+g‘𝑊) |
mapdindp1.o | ⊢ 0 = (0g‘𝑊) |
mapdindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
mapdindp1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
mapdindp1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.W | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.e | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
mapdindp1.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdindp1.f | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
mapdindp1.yz | ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) |
Ref | Expression |
---|---|
mapdindp0 | ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | mapdindp1.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | mapdindp1.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 21080 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | mapdindp1.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
7 | 6 | eldifad 3958 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
8 | mapdindp1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
9 | 8, 1, 2 | lspsncl 20950 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
10 | 5, 7, 9 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
11 | mapdindp1.e | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
12 | 11, 10 | eqeltrrd 2827 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) |
13 | eqid 2726 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
14 | 1, 13 | lsmcl 21057 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊)) |
15 | 5, 10, 12, 14 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊)) |
16 | 1 | lsssssubg 20931 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
17 | 5, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
18 | 17, 10 | sseldd 3979 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
19 | 11, 18 | eqeltrrd 2827 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) |
20 | 8, 2 | lspsnid 20966 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌})) |
21 | 5, 7, 20 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌})) |
22 | mapdindp1.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
23 | 22 | eldifad 3958 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
24 | 8, 2 | lspsnid 20966 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍})) |
25 | 5, 23, 24 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍})) |
26 | mapdindp1.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
27 | 26, 13 | lsmelvali 19644 | . . . . 5 ⊢ ((((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) ∧ (𝑌 ∈ (𝑁‘{𝑌}) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
28 | 18, 19, 21, 25, 27 | syl22anc 837 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
29 | 1, 2, 5, 15, 28 | ellspsn5 20969 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
30 | 11 | oveq2d 7432 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍}))) |
31 | 13 | lsmidm 19657 | . . . . 5 ⊢ ((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌})) |
32 | 18, 31 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌})) |
33 | 30, 32 | eqtr3d 2768 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) = (𝑁‘{𝑌})) |
34 | 29, 33 | sseqtrd 4019 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌})) |
35 | mapdindp1.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
36 | 8, 26 | lmodvacl 20847 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
37 | 5, 7, 23, 36 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
38 | mapdindp1.yz | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) | |
39 | eldifsn 4785 | . . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 )) | |
40 | 37, 38, 39 | sylanbrc 581 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 })) |
41 | 8, 35, 2, 3, 40, 7 | lspsncmp 21093 | . 2 ⊢ (𝜑 → ((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))) |
42 | 34, 41 | mpbid 231 | 1 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3943 ⊆ wss 3946 {csn 4623 {cpr 4625 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +gcplusg 17261 0gc0g 17449 SubGrpcsubg 19110 LSSumclsm 19628 LModclmod 20832 LSubSpclss 20904 LSpanclspn 20944 LVecclvec 21076 |
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 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-cntz 19307 df-lsm 19630 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-drng 20705 df-lmod 20834 df-lss 20905 df-lsp 20945 df-lvec 21077 |
This theorem is referenced by: mapdindp1 41432 mapdindp2 41433 |
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