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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelsubginvcld | Structured version Visualization version GIF version |
Description: The inverse of a non-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) |
nelsubginvcld.p | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
nelsubginvcld | ⊢ (𝜑 → (𝑁‘𝑋) ∈ (𝐵 ∖ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelsubginvcld.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | nelsubginvcld.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) | |
3 | 2 | eldifad 3955 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
4 | nelsubginvcld.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | nelsubginvcld.p | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 4, 5 | grpinvcl 18846 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
7 | 1, 3, 6 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
8 | 2 | eldifbd 3956 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) |
9 | 4, 5 | grpinvinv 18863 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
10 | 1, 3, 9 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
11 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) ∈ 𝑆) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
12 | nelsubginvcld.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
13 | 5 | subginvcl 18986 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑁‘𝑋) ∈ 𝑆) → (𝑁‘(𝑁‘𝑋)) ∈ 𝑆) |
14 | 12, 13 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) ∈ 𝑆) → (𝑁‘(𝑁‘𝑋)) ∈ 𝑆) |
15 | 11, 14 | eqeltrrd 2833 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
16 | 8, 15 | mtand 814 | . 2 ⊢ (𝜑 → ¬ (𝑁‘𝑋) ∈ 𝑆) |
17 | 7, 16 | eldifd 3954 | 1 ⊢ (𝜑 → (𝑁‘𝑋) ∈ (𝐵 ∖ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3940 ‘cfv 6531 Basecbs 17125 Grpcgrp 18793 invgcminusg 18794 SubGrpcsubg 18971 |
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 2702 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7838 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-er 8685 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-nn 12194 df-2 12256 df-sets 17078 df-slot 17096 df-ndx 17108 df-base 17126 df-ress 17155 df-plusg 17191 df-0g 17368 df-mgm 18542 df-sgrp 18591 df-mnd 18602 df-grp 18796 df-minusg 18797 df-subg 18974 |
This theorem is referenced by: (None) |
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