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Mirrors > Home > MPE Home > Th. List > subgid | Structured version Visualization version GIF version |
Description: A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
issubg.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
subgid | ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
2 | ssidd 3989 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ⊆ 𝐵) | |
3 | issubg.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | ressid 16549 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = 𝐺) |
5 | 4, 1 | eqeltrd 2913 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp) |
6 | 3 | issubg 18219 | . 2 ⊢ (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵 ⊆ 𝐵 ∧ (𝐺 ↾s 𝐵) ∈ Grp)) |
7 | 1, 2, 5, 6 | syl3anbrc 1335 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 ↾s cress 16474 Grpcgrp 18043 SubGrpcsubg 18213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-ress 16481 df-subg 18216 |
This theorem is referenced by: trivsubgsnd 18246 nsgid 18262 gaid2 18373 pgpfac1 19133 pgpfac 19137 ablfaclem2 19139 ablfac 19141 qusxpid 30856 |
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