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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatexch | Structured version Visualization version GIF version |
Description: The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 30161 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatexch.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatexch.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatexch.o | ⊢ 0 = (0g‘𝑊) |
lsatexch.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatexch.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatexch.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatexch.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatexch.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatexch.l | ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) |
lsatexch.z | ⊢ (𝜑 → (𝑈 ∩ 𝑄) = { 0 }) |
Ref | Expression |
---|---|
lsatexch | ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatexch.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lveclmod 19881 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | lsatexch.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | 4 | lsssssubg 19733 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
7 | lsatexch.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 6, 7 | sseldd 3971 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
9 | lsatexch.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
10 | lsatexch.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
11 | 4, 9, 3, 10 | lsatlssel 36137 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
12 | 6, 11 | sseldd 3971 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
13 | lsatexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
14 | 13 | lsmub2 18786 | . . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑈 ⊕ 𝑅)) |
15 | 8, 12, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑅)) |
16 | eqid 2824 | . . 3 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
17 | 4, 13 | lsmcl 19858 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
18 | 3, 7, 11, 17 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
19 | lsatexch.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
20 | 4, 9, 3, 19 | lsatlssel 36137 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
21 | 4, 13 | lsmcl 19858 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆) → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
22 | 3, 7, 20, 21 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
23 | lsatexch.z | . . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ 𝑄) = { 0 }) | |
24 | lsatexch.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑊) | |
25 | 4, 13, 24, 9, 16, 1, 7, 19 | lcvp 36180 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ 𝑄) = { 0 } ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑄))) |
26 | 23, 25 | mpbid 234 | . . . . . 6 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑄)) |
27 | 4, 16, 1, 7, 22, 26 | lcvpss 36164 | . . . . 5 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ 𝑄)) |
28 | 13 | lsmub1 18785 | . . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (𝑈 ⊕ 𝑅)) |
29 | 8, 12, 28 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑈 ⊕ 𝑅)) |
30 | lsatexch.l | . . . . . 6 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
31 | 6, 20 | sseldd 3971 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 6, 18 | sseldd 3971 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
33 | 13 | lsmlub 18793 | . . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) → ((𝑈 ⊆ (𝑈 ⊕ 𝑅) ∧ 𝑄 ⊆ (𝑈 ⊕ 𝑅)) ↔ (𝑈 ⊕ 𝑄) ⊆ (𝑈 ⊕ 𝑅))) |
34 | 8, 31, 32, 33 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → ((𝑈 ⊆ (𝑈 ⊕ 𝑅) ∧ 𝑄 ⊆ (𝑈 ⊕ 𝑅)) ↔ (𝑈 ⊕ 𝑄) ⊆ (𝑈 ⊕ 𝑅))) |
35 | 29, 30, 34 | mpbi2and 710 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) ⊆ (𝑈 ⊕ 𝑅)) |
36 | 27, 35 | psssstrd 4089 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ 𝑅)) |
37 | 4, 13, 9, 16, 1, 7, 10 | lcv2 36182 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑈 ⊕ 𝑅) ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) |
38 | 36, 37 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) |
39 | 4, 16, 1, 7, 18, 22, 38, 27, 35 | lcvnbtwn2 36167 | . 2 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = (𝑈 ⊕ 𝑅)) |
40 | 15, 39 | sseqtrrd 4011 | 1 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ⊆ wss 3939 ⊊ wpss 3940 {csn 4570 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 0gc0g 16716 SubGrpcsubg 18276 LSSumclsm 18762 LModclmod 19637 LSubSpclss 19706 LVecclvec 19877 LSAtomsclsa 36114 ⋖L clcv 36158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lcv 36159 |
This theorem is referenced by: lsatexch1 36186 |
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