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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 3950 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
4 | nelsubginvcld.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
5 | nelsubginvcld.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
6 | 5 | subgss 18282 | . . . . 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 3970 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
10 | nelsubgcld.p | . . . 4 ⊢ + = (+g‘𝐺) | |
11 | 5, 10 | grpcl 18113 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
12 | 1, 3, 9, 11 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
13 | 2 | eldifbd 3951 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) |
14 | eqid 2823 | . . . . . . 7 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
15 | 5, 10, 14 | grppncan 18192 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) = 𝑋) |
16 | 1, 3, 9, 15 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) = 𝑋) |
17 | 16 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) = 𝑋) |
18 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺)) |
19 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | |
20 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → 𝑌 ∈ 𝑆) |
21 | 14 | subgsubcl 18292 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 + 𝑌) ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) ∈ 𝑆) |
22 | 18, 19, 20, 21 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → ((𝑋 + 𝑌)(-g‘𝐺)𝑌) ∈ 𝑆) |
23 | 17, 22 | eqeltrrd 2916 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
24 | 13, 23 | mtand 814 | . 2 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑆) |
25 | 12, 24 | eldifd 3949 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Grpcgrp 18105 -gcsg 18107 SubGrpcsubg 18275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 |
This theorem is referenced by: nelsubgsubcld 39137 |
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