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Theorem subgid 17804
Description: A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypothesis
Ref Expression
issubg.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
subgid (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))

Proof of Theorem subgid
StepHypRef Expression
1 id 22 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Grp)
2 ssid 3773 . . 3 𝐵𝐵
32a1i 11 . 2 (𝐺 ∈ Grp → 𝐵𝐵)
4 issubg.b . . . 4 𝐵 = (Base‘𝐺)
54ressid 16142 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) = 𝐺)
65, 1eqeltrd 2850 . 2 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
74issubg 17802 . 2 (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵𝐵 ∧ (𝐺s 𝐵) ∈ Grp))
81, 3, 6, 7syl3anbrc 1428 1 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wss 3723  cfv 6031  (class class class)co 6793  Basecbs 16064  s cress 16065  Grpcgrp 17630  SubGrpcsubg 17796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-ress 16072  df-subg 17799
This theorem is referenced by:  nsgid  17848  gaid2  17943  pgpfac1  18687  pgpfac  18691  ablfaclem2  18693  ablfac  18695
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