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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelsubgsubcld | Structured version Visualization version GIF version |
Description: A non-subgroup-member minus 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) |
nelsubgsubcld.p | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
nelsubgsubcld | ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐵 ∖ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelsubginvcld.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆)) | |
2 | 1 | eldifad 3961 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
3 | nelsubginvcld.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
4 | nelsubginvcld.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 4 | subgss 19044 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
7 | nelsubgcld.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
8 | 6, 7 | sseldd 3984 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
9 | eqid 2731 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2731 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
11 | nelsubgsubcld.p | . . . 4 ⊢ − = (-g‘𝐺) | |
12 | 4, 9, 10, 11 | grpsubval 18907 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
13 | 2, 8, 12 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
14 | nelsubginvcld.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
15 | 10 | subginvcl 19052 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) ∈ 𝑆) |
16 | 3, 7, 15 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑌) ∈ 𝑆) |
17 | 14, 3, 1, 4, 16, 9 | nelsubgcld 41378 | . 2 ⊢ (𝜑 → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ (𝐵 ∖ 𝑆)) |
18 | 13, 17 | eqeltrd 2832 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐵 ∖ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∖ cdif 3946 ⊆ wss 3949 ‘cfv 6544 (class class class)co 7412 Basecbs 17149 +gcplusg 17202 Grpcgrp 18856 invgcminusg 18857 -gcsg 18858 SubGrpcsubg 19037 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 |
This theorem is referenced by: (None) |
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