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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 2823 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
2 | lspabs2.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lspabs2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19880 | . . . . . 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 19751 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
9 | 5, 6, 8 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
10 | lspabs3.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | 7, 1, 2 | lspsncl 19751 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
12 | 5, 10, 11 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
13 | eqid 2823 | . . . . . . 7 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
14 | 1, 13 | lsmcl 19857 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑊)) |
16 | 7, 2 | lspsnsubg 19754 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
17 | 5, 6, 16 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
18 | lspabs3.e | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | |
19 | 18, 17 | eqeltrrd 2916 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
20 | 7, 2 | lspsnid 19767 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
21 | 5, 6, 20 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
22 | 7, 2 | lspsnid 19767 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌})) |
23 | 5, 10, 22 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌})) |
24 | lspabs2.p | . . . . . . 7 ⊢ + = (+g‘𝑊) | |
25 | 24, 13 | lsmelvali 18777 | . . . . . 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 836 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
27 | 1, 2, 5, 15, 26 | lspsnel5a 19770 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
28 | 18 | oveq2d 7174 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
29 | 13 | lsmidm 18790 | . . . . . 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 2860 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋})) |
32 | 27, 31 | sseqtrd 4009 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋})) |
33 | lspabs2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
34 | 7, 24 | lmodvacl 19650 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
35 | 5, 6, 10, 34 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
36 | lspabs3.xy | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) | |
37 | eldifsn 4721 | . . . . 5 ⊢ ((𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑋 + 𝑌) ∈ 𝑉 ∧ (𝑋 + 𝑌) ≠ 0 )) | |
38 | 35, 36, 37 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
39 | 7, 33, 2, 3, 38, 6 | lspsncmp 19890 | . . 3 ⊢ (𝜑 → ((𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{(𝑋 + 𝑌)}) = (𝑁‘{𝑋}))) |
40 | 32, 39 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = (𝑁‘{𝑋})) |
41 | 40 | eqcomd 2829 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 SubGrpcsubg 18275 LSSumclsm 18761 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 |
This theorem is referenced by: (None) |
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