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Mirrors > Home > ILE Home > Th. List > subg0 | GIF version |
Description: A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
subg0.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
subg0.i | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
subg0 | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subg0.h | . . . . . 6 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | 1 | a1i 9 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆)) |
3 | eqid 2177 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺)) |
5 | id 19 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
6 | subgrcl 12970 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
7 | 2, 4, 5, 6 | ressplusgd 12579 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻)) |
8 | 7 | oveqd 5889 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻))) |
9 | 1 | subggrp 12968 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
10 | eqid 2177 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
11 | eqid 2177 | . . . . . 6 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
12 | 10, 11 | grpidcl 12836 | . . . . 5 ⊢ (𝐻 ∈ Grp → (0g‘𝐻) ∈ (Base‘𝐻)) |
13 | 9, 12 | syl 14 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ (Base‘𝐻)) |
14 | eqid 2177 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
15 | 10, 14, 11 | grplid 12838 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ (0g‘𝐻) ∈ (Base‘𝐻)) → ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻)) = (0g‘𝐻)) |
16 | 9, 13, 15 | syl2anc 411 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻)) = (0g‘𝐻)) |
17 | 8, 16 | eqtrd 2210 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻)) |
18 | eqid 2177 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
19 | 18 | subgss 12965 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
20 | 1 | subgbas 12969 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
21 | 13, 20 | eleqtrrd 2257 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ 𝑆) |
22 | 19, 21 | sseldd 3156 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ (Base‘𝐺)) |
23 | subg0.i | . . . 4 ⊢ 0 = (0g‘𝐺) | |
24 | 18, 3, 23 | grpid 12844 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (0g‘𝐻) ∈ (Base‘𝐺)) → (((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻) ↔ 0 = (0g‘𝐻))) |
25 | 6, 22, 24 | syl2anc 411 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻) ↔ 0 = (0g‘𝐻))) |
26 | 17, 25 | mpbid 147 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 ↾s cress 12455 +gcplusg 12528 0gc0g 12693 Grpcgrp 12809 SubGrpcsubg 12958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-subg 12961 |
This theorem is referenced by: subginv 12972 subg0cl 12973 subgmulg 12979 subrg0 13287 |
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