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Mathbox for Norm Megill |
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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 32410 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 21123 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | lsatexch.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | 4 | lsssssubg 20974 | . . . . 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 3996 | . . 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 38979 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
12 | 6, 11 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
13 | lsatexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
14 | 13 | lsmub2 19691 | . . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑈 ⊕ 𝑅)) |
15 | 8, 12, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑅)) |
16 | eqid 2735 | . . 3 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
17 | 4, 13 | lsmcl 21100 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
18 | 3, 7, 11, 17 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
19 | lsatexch.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
20 | 4, 9, 3, 19 | lsatlssel 38979 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
21 | 4, 13 | lsmcl 21100 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆) → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
22 | 3, 7, 20, 21 | syl3anc 1370 | . . 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 39022 | . . . . . . 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 39006 | . . . . 5 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ 𝑄)) |
28 | 13 | lsmub1 19690 | . . . . . . 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 3996 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 6, 18 | sseldd 3996 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
33 | 13 | lsmlub 19697 | . . . . . . 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 1370 | . . . . . 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 4122 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ 𝑅)) |
37 | 4, 13, 9, 16, 1, 7, 10 | lcv2 39024 | . . . 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 39009 | . 2 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = (𝑈 ⊕ 𝑅)) |
40 | 15, 39 | sseqtrrd 4037 | 1 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 ⊊ wpss 3964 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0gc0g 17486 SubGrpcsubg 19151 LSSumclsm 19667 LModclmod 20875 LSubSpclss 20947 LVecclvec 21119 LSAtomsclsa 38956 ⋖L clcv 39000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lcv 39001 |
This theorem is referenced by: lsatexch1 39028 |
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