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Mirrors > Home > MPE Home > Th. List > lsmidm | Structured version Visualization version GIF version |
Description: Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
Ref | Expression |
---|---|
lsmub1.p | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmidm | ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2826 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2826 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | lsmub1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
4 | 1, 2, 3 | lsmval 18415 | . . . 4 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑈 ⊕ 𝑈) = ran (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦))) |
5 | 4 | anidms 564 | . . 3 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = ran (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦))) |
6 | 2 | subgcl 17956 | . . . . . . 7 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑈) |
7 | 6 | 3expb 1155 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑈) |
8 | 7 | ralrimivva 3181 | . . . . 5 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑥(+g‘𝐺)𝑦) ∈ 𝑈) |
9 | eqid 2826 | . . . . . 6 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦)) | |
10 | 9 | fmpt2 7501 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑥(+g‘𝐺)𝑦) ∈ 𝑈 ↔ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦)):(𝑈 × 𝑈)⟶𝑈) |
11 | 8, 10 | sylib 210 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦)):(𝑈 × 𝑈)⟶𝑈) |
12 | 11 | frnd 6286 | . . 3 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ 𝑈) |
13 | 5, 12 | eqsstrd 3865 | . 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) ⊆ 𝑈) |
14 | 3 | lsmub1 18423 | . . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (𝑈 ⊕ 𝑈)) |
15 | 14 | anidms 564 | . 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (𝑈 ⊕ 𝑈)) |
16 | 13, 15 | eqssd 3845 | 1 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ⊆ wss 3799 × cxp 5341 ran crn 5344 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 ↦ cmpt2 6908 Basecbs 16223 +gcplusg 16306 SubGrpcsubg 17940 LSSumclsm 18401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-subg 17943 df-lsm 18403 |
This theorem is referenced by: lsmlub 18430 lspabs2 19480 lspabs3 19481 lsatcv0eq 35123 lsatcv1 35124 lsatcvat3 35128 dia2dimlem13 37152 dihjatcclem1 37494 dvh3dimatN 37515 dvh2dimatN 37516 mapdindp0 37795 mapdh6dN 37815 hdmap1l6d 37889 |
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