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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 18927 | . . . . . . 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 18801 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
8 | 3, 4, 7 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
9 | lspabs2.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
10 | 9 | eldifad 3552 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
11 | 5, 6 | lspsnsubg 18801 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
12 | 3, 10, 11 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) |
13 | eqid 2610 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
14 | 13 | lsmub2 17895 | . . . . 5 ⊢ (((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
15 | 8, 12, 14 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
16 | lspabs2.e | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) | |
17 | 16 | oveq2d 6565 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)}))) |
18 | 13 | lsmidm 17900 | . . . . . 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 18916 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌})) |
22 | 5, 20 | lmodvacl 18700 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
23 | 3, 4, 10, 22 | syl3anc 1318 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
24 | 5, 6, 13, 3, 4, 23 | lsmpr 18910 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)}))) |
25 | 5, 6, 13, 3, 4, 10 | lsmpr 18910 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
26 | 21, 24, 25 | 3eqtr3d 2652 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
27 | 17, 19, 26 | 3eqtr3rd 2653 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋})) |
28 | 15, 27 | sseqtrd 3604 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋})) |
29 | lspabs2.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
30 | 5, 29, 6, 1, 9, 4 | lspsncmp 18937 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑌}) = (𝑁‘{𝑋}))) |
31 | 28, 30 | mpbid 221 | . 2 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋})) |
32 | 31 | eqcomd 2616 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 {cpr 4127 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 SubGrpcsubg 17411 LSSumclsm 17872 LModclmod 18686 LSpanclspn 18792 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 |
This theorem is referenced by: lspindp3 18957 |
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