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Mirrors > Home > MPE Home > Th. List > lsmmod2 | Structured version Visualization version GIF version |
Description: Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
lsmmod.p | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmmod2 | ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1186 | . . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘𝐺)) | |
2 | eqid 2797 | . . . . . . 7 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
3 | 2 | oppgsubg 18236 | . . . . . 6 ⊢ (SubGrp‘𝐺) = (SubGrp‘(oppg‘𝐺)) |
4 | 1, 3 | syl6eleq 2895 | . . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ∈ (SubGrp‘(oppg‘𝐺))) |
5 | simpl2 1185 | . . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘𝐺)) | |
6 | 5, 3 | syl6eleq 2895 | . . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑇 ∈ (SubGrp‘(oppg‘𝐺))) |
7 | simpl1 1184 | . . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺)) | |
8 | 7, 3 | syl6eleq 2895 | . . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) |
9 | simpr 485 | . . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → 𝑈 ⊆ 𝑆) | |
10 | eqid 2797 | . . . . . 6 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
11 | 10 | lsmmod 18532 | . . . . 5 ⊢ (((𝑈 ∈ (SubGrp‘(oppg‘𝐺)) ∧ 𝑇 ∈ (SubGrp‘(oppg‘𝐺)) ∧ 𝑆 ∈ (SubGrp‘(oppg‘𝐺))) ∧ 𝑈 ⊆ 𝑆) → (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆)) |
12 | 4, 6, 8, 9, 11 | syl31anc 1366 | . . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆)) |
13 | 12 | eqcomd 2803 | . . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆))) |
14 | incom 4105 | . . 3 ⊢ ((𝑈(LSSum‘(oppg‘𝐺))𝑇) ∩ 𝑆) = (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) | |
15 | incom 4105 | . . . 4 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇) | |
16 | 15 | oveq2i 7034 | . . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑇 ∩ 𝑆)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)) |
17 | 13, 14, 16 | 3eqtr3g 2856 | . 2 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇))) |
18 | lsmmod.p | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
19 | 2, 18 | oppglsm 18501 | . . 3 ⊢ (𝑈(LSSum‘(oppg‘𝐺))𝑇) = (𝑇 ⊕ 𝑈) |
20 | 19 | ineq2i 4112 | . 2 ⊢ (𝑆 ∩ (𝑈(LSSum‘(oppg‘𝐺))𝑇)) = (𝑆 ∩ (𝑇 ⊕ 𝑈)) |
21 | 2, 18 | oppglsm 18501 | . 2 ⊢ (𝑈(LSSum‘(oppg‘𝐺))(𝑆 ∩ 𝑇)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈) |
22 | 17, 20, 21 | 3eqtr3g 2856 | 1 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∩ cin 3864 ⊆ wss 3865 ‘cfv 6232 (class class class)co 7023 SubGrpcsubg 18031 oppgcoppg 18218 LSSumclsm 18493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-0g 16548 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-subg 18034 df-oppg 18219 df-lsm 18495 |
This theorem is referenced by: lcvexchlem3 35724 lcfrlem23 38253 |
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