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Mirrors > Home > MPE Home > Th. List > 0subg | Structured version Visualization version GIF version |
Description: The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) |
Ref | Expression |
---|---|
0subg.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
0subg | ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subg.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 17273 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 4281 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ⊆ (Base‘𝐺)) |
5 | fvex 6113 | . . . . 5 ⊢ (0g‘𝐺) ∈ V | |
6 | 2, 5 | eqeltri 2684 | . . . 4 ⊢ 0 ∈ V |
7 | 6 | snnz 4252 | . . 3 ⊢ { 0 } ≠ ∅ |
8 | 7 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ≠ ∅) |
9 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | 1, 9, 2 | grplid 17275 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
11 | 3, 10 | mpdan 699 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
12 | ovex 6577 | . . . . 5 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V | |
13 | 12 | elsn 4140 | . . . 4 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | 11, 13 | sylibr 223 | . . 3 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
15 | eqid 2610 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
16 | 2, 15 | grpinvid 17299 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
17 | fvex 6113 | . . . . 5 ⊢ ((invg‘𝐺)‘ 0 ) ∈ V | |
18 | 17 | elsn 4140 | . . . 4 ⊢ (((invg‘𝐺)‘ 0 ) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) = 0 ) |
19 | 16, 18 | sylibr 223 | . . 3 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) ∈ { 0 }) |
20 | oveq1 6556 | . . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺)𝑏)) | |
21 | 20 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑎 = 0 → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
22 | 21 | ralbidv 2969 | . . . . . 6 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
23 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑏 = 0 → ( 0 (+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
24 | 23 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑏 = 0 → (( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
25 | 6, 24 | ralsn 4169 | . . . . . 6 ⊢ (∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
26 | 22, 25 | syl6bb 275 | . . . . 5 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
27 | fveq2 6103 | . . . . . 6 ⊢ (𝑎 = 0 → ((invg‘𝐺)‘𝑎) = ((invg‘𝐺)‘ 0 )) | |
28 | 27 | eleq1d 2672 | . . . . 5 ⊢ (𝑎 = 0 → (((invg‘𝐺)‘𝑎) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
29 | 26, 28 | anbi12d 743 | . . . 4 ⊢ (𝑎 = 0 → ((∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 }) ↔ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ∧ ((invg‘𝐺)‘ 0 ) ∈ { 0 }))) |
30 | 6, 29 | ralsn 4169 | . . 3 ⊢ (∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 }) ↔ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ∧ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
31 | 14, 19, 30 | sylanbrc 695 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })) |
32 | 1, 9, 15 | issubg2 17432 | . 2 ⊢ (𝐺 ∈ Grp → ({ 0 } ∈ (SubGrp‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ { 0 } ≠ ∅ ∧ ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })))) |
33 | 4, 8, 31, 32 | mpbir3and 1238 | 1 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 |
This theorem is referenced by: 0nsg 17462 idressubgsymg 17653 pgp0 17834 slwn0 17853 lsm01 17907 lsm02 17908 dprdz 18252 dprdsn 18258 pgpfac1lem5 18301 tgptsmscls 21763 |
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