Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lsmdisjr | Structured version Visualization version GIF version |
Description: Disjointness from 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) |
lsmdisjr.i | ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) |
Ref | Expression |
---|---|
lsmdisjr | ⊢ (𝜑 → ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcntz.p | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
2 | lsmcntz.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
3 | lsmcntz.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
4 | lsmcntz.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
5 | lsmdisj.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
6 | incom 4115 | . . . 4 ⊢ (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑇 ⊕ 𝑈) ∩ 𝑆) | |
7 | lsmdisjr.i | . . . 4 ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | |
8 | 6, 7 | eqtr3id 2792 | . . 3 ⊢ (𝜑 → ((𝑇 ⊕ 𝑈) ∩ 𝑆) = { 0 }) |
9 | 1, 2, 3, 4, 5, 8 | lsmdisj 19071 | . 2 ⊢ (𝜑 → ((𝑇 ∩ 𝑆) = { 0 } ∧ (𝑈 ∩ 𝑆) = { 0 })) |
10 | incom 4115 | . . . 4 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇) | |
11 | 10 | eqeq1i 2742 | . . 3 ⊢ ((𝑇 ∩ 𝑆) = { 0 } ↔ (𝑆 ∩ 𝑇) = { 0 }) |
12 | incom 4115 | . . . 4 ⊢ (𝑈 ∩ 𝑆) = (𝑆 ∩ 𝑈) | |
13 | 12 | eqeq1i 2742 | . . 3 ⊢ ((𝑈 ∩ 𝑆) = { 0 } ↔ (𝑆 ∩ 𝑈) = { 0 }) |
14 | 11, 13 | anbi12i 630 | . 2 ⊢ (((𝑇 ∩ 𝑆) = { 0 } ∧ (𝑈 ∩ 𝑆) = { 0 }) ↔ ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) |
15 | 9, 14 | sylib 221 | 1 ⊢ (𝜑 → ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 {csn 4541 ‘cfv 6380 (class class class)co 7213 0gc0g 16944 SubGrpcsubg 18537 LSSumclsm 19023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-subg 18540 df-lsm 19025 |
This theorem is referenced by: lsmdisj2a 19077 lsmdisj2b 19078 |
Copyright terms: Public domain | W3C validator |