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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 2823 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subg.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18133 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 4744 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ⊆ (Base‘𝐺)) |
5 | 2 | fvexi 6686 | . . . 4 ⊢ 0 ∈ V |
6 | 5 | snnz 4713 | . . 3 ⊢ { 0 } ≠ ∅ |
7 | 6 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ≠ ∅) |
8 | eqid 2823 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | 1, 8, 2 | grplid 18135 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
10 | 3, 9 | mpdan 685 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
11 | ovex 7191 | . . . . 5 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V | |
12 | 11 | elsn 4584 | . . . 4 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 10, 12 | sylibr 236 | . . 3 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
14 | eqid 2823 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
15 | 2, 14 | grpinvid 18162 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
16 | fvex 6685 | . . . . 5 ⊢ ((invg‘𝐺)‘ 0 ) ∈ V | |
17 | 16 | elsn 4584 | . . . 4 ⊢ (((invg‘𝐺)‘ 0 ) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) = 0 ) |
18 | 15, 17 | sylibr 236 | . . 3 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) ∈ { 0 }) |
19 | oveq1 7165 | . . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺)𝑏)) | |
20 | 19 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑎 = 0 → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
21 | 20 | ralbidv 3199 | . . . . . 6 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
22 | oveq2 7166 | . . . . . . . 8 ⊢ (𝑏 = 0 → ( 0 (+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
23 | 22 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑏 = 0 → (( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
24 | 5, 23 | ralsn 4621 | . . . . . 6 ⊢ (∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
25 | 21, 24 | syl6bb 289 | . . . . 5 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
26 | fveq2 6672 | . . . . . 6 ⊢ (𝑎 = 0 → ((invg‘𝐺)‘𝑎) = ((invg‘𝐺)‘ 0 )) | |
27 | 26 | eleq1d 2899 | . . . . 5 ⊢ (𝑎 = 0 → (((invg‘𝐺)‘𝑎) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
28 | 25, 27 | anbi12d 632 | . . . 4 ⊢ (𝑎 = 0 → ((∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 }) ↔ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ∧ ((invg‘𝐺)‘ 0 ) ∈ { 0 }))) |
29 | 5, 28 | ralsn 4621 | . . 3 ⊢ (∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 }) ↔ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ∧ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
30 | 13, 18, 29 | sylanbrc 585 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })) |
31 | 1, 8, 14 | issubg2 18296 | . 2 ⊢ (𝐺 ∈ Grp → ({ 0 } ∈ (SubGrp‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ { 0 } ≠ ∅ ∧ ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })))) |
32 | 4, 7, 30, 31 | mpbir3and 1338 | 1 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⊆ wss 3938 ∅c0 4293 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Grpcgrp 18105 invgcminusg 18106 SubGrpcsubg 18275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 |
This theorem is referenced by: 0nsg 18323 idressubgsymg 18540 pgp0 18723 slwn0 18742 lsm01 18799 lsm02 18800 dprdz 19154 dprdsn 19160 pgpfac1lem5 19203 tgptsmscls 22760 evpmsubg 30791 |
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