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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 3624 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ⊆ 𝐵) |
4 | issubg.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 4 | ressid 15935 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = 𝐺) |
6 | 5, 1 | eqeltrd 2701 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp) |
7 | 4 | issubg 17594 | . 2 ⊢ (𝐵 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐵 ⊆ 𝐵 ∧ (𝐺 ↾s 𝐵) ∈ Grp)) |
8 | 1, 3, 6, 7 | syl3anbrc 1246 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 Grpcgrp 17422 SubGrpcsubg 17588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-ress 15865 df-subg 17591 |
This theorem is referenced by: nsgid 17640 gaid2 17736 pgpfac1 18479 pgpfac 18483 ablfaclem2 18485 ablfac 18487 |
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