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Mirrors > Home > MPE Home > Th. List > lspabs2 | Structured version Visualization version GIF version |
Description: Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.) |
Ref | Expression |
---|---|
lspabs2.v | ⊢ 𝑉 = (Base‘𝑊) |
lspabs2.p | ⊢ + = (+g‘𝑊) |
lspabs2.o | ⊢ 0 = (0g‘𝑊) |
lspabs2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspabs2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspabs2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspabs2.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lspabs2.e | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) |
Ref | Expression |
---|---|
lspabs2 | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspabs2.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lveclmod 20474 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | lspabs2.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
5 | lspabs2.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lspabs2.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊) | |
7 | 5, 6 | lspsnsubg 20348 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
8 | 3, 4, 7 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
9 | lspabs2.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
10 | 9 | eldifad 3910 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
11 | 5, 6 | lspsnsubg 20348 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
12 | 3, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
13 | eqid 2736 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
14 | 13 | lsmub2 19359 | . . . . 5 ⊢ (((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
15 | 8, 12, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
16 | lspabs2.e | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) | |
17 | 16 | oveq2d 7353 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)}))) |
18 | 13 | lsmidm 19364 | . . . . . 6 ⊢ ((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = (𝑁‘{𝑋})) |
19 | 8, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = (𝑁‘{𝑋})) |
20 | lspabs2.p | . . . . . . 7 ⊢ + = (+g‘𝑊) | |
21 | 5, 20, 6, 3, 4, 10 | lspprabs 20463 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌})) |
22 | 5, 20 | lmodvacl 20243 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
23 | 3, 4, 10, 22 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
24 | 5, 6, 13, 3, 4, 23 | lsmpr 20457 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)}))) |
25 | 5, 6, 13, 3, 4, 10 | lsmpr 20457 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
26 | 21, 24, 25 | 3eqtr3d 2784 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
27 | 17, 19, 26 | 3eqtr3rd 2785 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋})) |
28 | 15, 27 | sseqtrd 3972 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋})) |
29 | lspabs2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
30 | 5, 29, 6, 1, 9, 4 | lspsncmp 20484 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑌}) = (𝑁‘{𝑋}))) |
31 | 28, 30 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋})) |
32 | 31 | eqcomd 2742 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∖ cdif 3895 ⊆ wss 3898 {csn 4573 {cpr 4575 ‘cfv 6479 (class class class)co 7337 Basecbs 17009 +gcplusg 17059 0gc0g 17247 SubGrpcsubg 18845 LSSumclsm 19335 LModclmod 20229 LSpanclspn 20339 LVecclvec 20470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-tpos 8112 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-grp 18676 df-minusg 18677 df-sbg 18678 df-subg 18848 df-cntz 19019 df-lsm 19337 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-drng 20095 df-lmod 20231 df-lss 20300 df-lsp 20340 df-lvec 20471 |
This theorem is referenced by: lspindp3 20504 |
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