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Mirrors > Home > MPE Home > Th. List > lspabs3 | 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) |
lspabs3.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspabs3.xy | ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) |
lspabs3.e | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lspabs3 | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | lspabs2.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lspabs2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19807 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | lspabs2.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
7 | lspabs2.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
8 | 7, 1, 2 | lspsncl 19678 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
9 | 5, 6, 8 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
10 | lspabs3.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | 7, 1, 2 | lspsncl 19678 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
12 | 5, 10, 11 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
13 | eqid 2818 | . . . . . . 7 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
14 | 1, 13 | lsmcl 19784 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑊)) |
15 | 5, 9, 12, 14 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑊)) |
16 | 7, 2 | lspsnsubg 19681 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
17 | 5, 6, 16 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
18 | lspabs3.e | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | |
19 | 18, 17 | eqeltrrd 2911 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
20 | 7, 2 | lspsnid 19694 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
21 | 5, 6, 20 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
22 | 7, 2 | lspsnid 19694 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌})) |
23 | 5, 10, 22 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌})) |
24 | lspabs2.p | . . . . . . 7 ⊢ + = (+g‘𝑊) | |
25 | 24, 13 | lsmelvali 18704 | . . . . . 6 ⊢ ((((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) ∧ (𝑋 ∈ (𝑁‘{𝑋}) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → (𝑋 + 𝑌) ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
26 | 17, 19, 21, 23, 25 | syl22anc 834 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
27 | 1, 2, 5, 15, 26 | lspsnel5a 19697 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
28 | 18 | oveq2d 7161 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
29 | 13 | lsmidm 18717 | . . . . . 6 ⊢ ((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = (𝑁‘{𝑋})) |
30 | 17, 29 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = (𝑁‘{𝑋})) |
31 | 28, 30 | eqtr3d 2855 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋})) |
32 | 27, 31 | sseqtrd 4004 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋})) |
33 | lspabs2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
34 | 7, 24 | lmodvacl 19577 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
35 | 5, 6, 10, 34 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
36 | lspabs3.xy | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) | |
37 | eldifsn 4711 | . . . . 5 ⊢ ((𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑋 + 𝑌) ∈ 𝑉 ∧ (𝑋 + 𝑌) ≠ 0 )) | |
38 | 35, 36, 37 | sylanbrc 583 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
39 | 7, 33, 2, 3, 38, 6 | lspsncmp 19817 | . . 3 ⊢ (𝜑 → ((𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{(𝑋 + 𝑌)}) = (𝑁‘{𝑋}))) |
40 | 32, 39 | mpbid 233 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = (𝑁‘{𝑋})) |
41 | 40 | eqcomd 2824 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 SubGrpcsubg 18211 LSSumclsm 18688 LModclmod 19563 LSubSpclss 19632 LSpanclspn 19672 LVecclvec 19803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 |
This theorem is referenced by: (None) |
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