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Mirrors > Home > ILE Home > Th. List > subgbas | GIF version |
Description: The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
subggrp.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
Ref | Expression |
---|---|
subgbas | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subggrp.h | . . 3 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | 1 | a1i 9 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆)) |
3 | eqid 2177 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 3 | a1i 9 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (Base‘𝐺) = (Base‘𝐺)) |
5 | 3 | issubg 12964 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
6 | 5 | simp1bi 1012 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
7 | 3 | subgss 12965 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
8 | 2, 4, 6, 7 | ressbas2d 12520 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 ↾s cress 12455 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-1re 7902 ax-addrcl 7905 |
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-ral 2460 df-rex 2461 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-ov 5875 df-oprab 5876 df-mpo 5877 df-inn 8916 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-subg 12961 |
This theorem is referenced by: subg0 12971 subginv 12972 subg0cl 12973 subginvcl 12974 subgcl 12975 subgsub 12977 subgmulg 12979 issubg2m 12980 subsubg 12988 nmznsg 13004 subrgbas 13289 issubrg2 13300 |
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