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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 32400 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 21105 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| 4 | lsatexch.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 5 | 4 | lsssssubg 20956 | . . . . 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 3984 | . . 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 38998 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | 
| 12 | 6, 11 | sseldd 3984 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) | 
| 13 | lsatexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
| 14 | 13 | lsmub2 19676 | . . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑈 ⊕ 𝑅)) | 
| 15 | 8, 12, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑅)) | 
| 16 | eqid 2737 | . . 3 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
| 17 | 4, 13 | lsmcl 21082 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) | 
| 18 | 3, 7, 11, 17 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) | 
| 19 | lsatexch.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
| 20 | 4, 9, 3, 19 | lsatlssel 38998 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) | 
| 21 | 4, 13 | lsmcl 21082 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆) → (𝑈 ⊕ 𝑄) ∈ 𝑆) | 
| 22 | 3, 7, 20, 21 | syl3anc 1373 | . . 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 39041 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ 𝑄) = { 0 } ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑄))) | 
| 26 | 23, 25 | mpbid 232 | . . . . . 6 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑄)) | 
| 27 | 4, 16, 1, 7, 22, 26 | lcvpss 39025 | . . . . 5 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ 𝑄)) | 
| 28 | 13 | lsmub1 19675 | . . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (𝑈 ⊕ 𝑅)) | 
| 29 | 8, 12, 28 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑈 ⊕ 𝑅)) | 
| 30 | lsatexch.l | . . . . . 6 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
| 31 | 6, 20 | sseldd 3984 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) | 
| 32 | 6, 18 | sseldd 3984 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) | 
| 33 | 13 | lsmlub 19682 | . . . . . . 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 1373 | . . . . . 6 ⊢ (𝜑 → ((𝑈 ⊆ (𝑈 ⊕ 𝑅) ∧ 𝑄 ⊆ (𝑈 ⊕ 𝑅)) ↔ (𝑈 ⊕ 𝑄) ⊆ (𝑈 ⊕ 𝑅))) | 
| 35 | 29, 30, 34 | mpbi2and 712 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) ⊆ (𝑈 ⊕ 𝑅)) | 
| 36 | 27, 35 | psssstrd 4112 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ 𝑅)) | 
| 37 | 4, 13, 9, 16, 1, 7, 10 | lcv2 39043 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑈 ⊕ 𝑅) ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) | 
| 38 | 36, 37 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) | 
| 39 | 4, 16, 1, 7, 18, 22, 38, 27, 35 | lcvnbtwn2 39028 | . 2 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = (𝑈 ⊕ 𝑅)) | 
| 40 | 15, 39 | sseqtrrd 4021 | 1 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ⊊ wpss 3952 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0gc0g 17484 SubGrpcsubg 19138 LSSumclsm 19652 LModclmod 20858 LSubSpclss 20929 LVecclvec 21101 LSAtomsclsa 38975 ⋖L clcv 39019 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lcv 39020 | 
| This theorem is referenced by: lsatexch1 39047 | 
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