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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 | ssid 3587 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ⊆ 𝐵) |
4 | issubg.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 4 | ressid 15762 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = 𝐺) |
6 | 5, 1 | eqeltrd 2688 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp) |
7 | 4 | issubg 17417 | . 2 ⊢ (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵 ⊆ 𝐵 ∧ (𝐺 ↾s 𝐵) ∈ Grp)) |
8 | 1, 3, 6, 7 | syl3anbrc 1239 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Grpcgrp 17245 SubGrpcsubg 17411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-ress 15702 df-subg 17414 |
This theorem is referenced by: nsgid 17463 gaid2 17559 pgpfac1 18302 pgpfac 18306 ablfaclem2 18308 ablfac 18310 |
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