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Mirrors > Home > ILE Home > Th. List > issubgrpd | GIF version |
Description: Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
Ref | Expression |
---|---|
issubgrpd.s | ⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) |
issubgrpd.z | ⊢ (𝜑 → 0 = (0g‘𝐼)) |
issubgrpd.p | ⊢ (𝜑 → + = (+g‘𝐼)) |
issubgrpd.ss | ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) |
issubgrpd.zcl | ⊢ (𝜑 → 0 ∈ 𝐷) |
issubgrpd.acl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) |
issubgrpd.ncl | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) |
issubgrpd.g | ⊢ (𝜑 → 𝐼 ∈ Grp) |
Ref | Expression |
---|---|
issubgrpd | ⊢ (𝜑 → 𝑆 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubgrpd.s | . 2 ⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) | |
2 | issubgrpd.z | . . . 4 ⊢ (𝜑 → 0 = (0g‘𝐼)) | |
3 | issubgrpd.p | . . . 4 ⊢ (𝜑 → + = (+g‘𝐼)) | |
4 | issubgrpd.ss | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) | |
5 | issubgrpd.zcl | . . . 4 ⊢ (𝜑 → 0 ∈ 𝐷) | |
6 | issubgrpd.acl | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) | |
7 | issubgrpd.ncl | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) | |
8 | issubgrpd.g | . . . 4 ⊢ (𝜑 → 𝐼 ∈ Grp) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | issubgrpd2 12981 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
10 | eqid 2177 | . . . 4 ⊢ (𝐼 ↾s 𝐷) = (𝐼 ↾s 𝐷) | |
11 | 10 | subggrp 12968 | . . 3 ⊢ (𝐷 ∈ (SubGrp‘𝐼) → (𝐼 ↾s 𝐷) ∈ Grp) |
12 | 9, 11 | syl 14 | . 2 ⊢ (𝜑 → (𝐼 ↾s 𝐷) ∈ Grp) |
13 | 1, 12 | eqeltrd 2254 | 1 ⊢ (𝜑 → 𝑆 ∈ Grp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 ↾s cress 12455 +gcplusg 12528 0gc0g 12693 Grpcgrp 12809 invgcminusg 12810 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-coll 4117 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-iun 3888 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-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 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-minusg 12813 df-subg 12961 |
This theorem is referenced by: (None) |
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