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Mirrors > Home > MPE Home > Th. List > lsmdisj3a | Structured version Visualization version GIF version |
Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
lsmcntz.p | ⊢ ⊕ = (LSSum‘𝐺) |
lsmcntz.s | ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) |
lsmcntz.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
lsmcntz.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
lsmdisj.o | ⊢ 0 = (0g‘𝐺) |
lsmdisj3b.z | ⊢ 𝑍 = (Cntz‘𝐺) |
lsmdisj3a.2 | ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) |
Ref | Expression |
---|---|
lsmdisj3a | ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcntz.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | lsmcntz.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
3 | lsmdisj3a.2 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) | |
4 | lsmcntz.p | . . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺) | |
5 | lsmdisj3b.z | . . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺) | |
6 | 4, 5 | lsmcom2 18070 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝑆 ⊕ 𝑇) = (𝑇 ⊕ 𝑆)) |
7 | 1, 2, 3, 6 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → (𝑆 ⊕ 𝑇) = (𝑇 ⊕ 𝑆)) |
8 | 7 | ineq1d 3813 | . . . 4 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑆) ∩ 𝑈)) |
9 | 8 | eqeq1d 2624 | . . 3 ⊢ (𝜑 → (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ↔ ((𝑇 ⊕ 𝑆) ∩ 𝑈) = { 0 })) |
10 | incom 3805 | . . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)) |
12 | 11 | eqeq1d 2624 | . . 3 ⊢ (𝜑 → ((𝑆 ∩ 𝑇) = { 0 } ↔ (𝑇 ∩ 𝑆) = { 0 })) |
13 | 9, 12 | anbi12d 747 | . 2 ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ (((𝑇 ⊕ 𝑆) ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑆) = { 0 }))) |
14 | lsmcntz.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
15 | lsmdisj.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
16 | 4, 2, 1, 14, 15 | lsmdisj2a 18100 | . 2 ⊢ (𝜑 → ((((𝑇 ⊕ 𝑆) ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑆) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))) |
17 | 13, 16 | bitrd 268 | 1 ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ‘cfv 5888 (class class class)co 6650 0gc0g 16100 SubGrpcsubg 17588 Cntzccntz 17748 LSSumclsm 18049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-subg 17591 df-cntz 17750 df-lsm 18051 |
This theorem is referenced by: (None) |
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