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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelsubgcld | Structured version Visualization version GIF version |
Description: A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.) |
Ref | Expression |
---|---|
nelsubginvcld.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
nelsubginvcld.s | ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) |
nelsubginvcld.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) |
nelsubginvcld.b | ⊢ 𝐵 = (Base‘𝐺) |
nelsubgcld.y | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
nelsubgcld.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
nelsubgcld | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelsubginvcld.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | nelsubginvcld.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) | |
3 | 2 | eldifad 3920 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
4 | nelsubginvcld.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
5 | nelsubginvcld.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
6 | 5 | subgss 18887 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
8 | nelsubgcld.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
9 | 7, 8 | sseldd 3943 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
10 | nelsubgcld.p | . . . 4 ⊢ + = (+g‘𝐺) | |
11 | 5, 10 | grpcl 18715 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
12 | 1, 3, 9, 11 | syl3anc 1371 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
13 | 2 | eldifbd 3921 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) |
14 | eqid 2737 | . . . . . . 7 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
15 | 5, 10, 14 | grppncan 18796 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) = 𝑋) |
16 | 1, 3, 9, 15 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) = 𝑋) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) = 𝑋) |
18 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺)) |
19 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | |
20 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → 𝑌 ∈ 𝑆) |
21 | 14 | subgsubcl 18897 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 + 𝑌) ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) ∈ 𝑆) |
22 | 18, 19, 20, 21 | syl3anc 1371 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) ∈ 𝑆) |
23 | 17, 22 | eqeltrrd 2839 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
24 | 13, 23 | mtand 814 | . 2 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑆) |
25 | 12, 24 | eldifd 3919 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 ⊆ wss 3908 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 +gcplusg 17092 Grpcgrp 18707 -gcsg 18709 SubGrpcsubg 18880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-minusg 18711 df-sbg 18712 df-subg 18883 |
This theorem is referenced by: nelsubgsubcld 40602 |
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