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Mirrors > Home > MPE Home > Th. List > issubgrpd | Structured version Visualization version 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 18761 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
10 | eqid 2740 | . . . 4 ⊢ (𝐼 ↾s 𝐷) = (𝐼 ↾s 𝐷) | |
11 | 10 | subggrp 18748 | . . 3 ⊢ (𝐷 ∈ (SubGrp‘𝐼) → (𝐼 ↾s 𝐷) ∈ Grp) |
12 | 9, 11 | syl 17 | . 2 ⊢ (𝜑 → (𝐼 ↾s 𝐷) ∈ Grp) |
13 | 1, 12 | eqeltrd 2841 | 1 ⊢ (𝜑 → 𝑆 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 ↾s cress 16931 +gcplusg 16952 0gc0g 17140 Grpcgrp 18567 invgcminusg 18568 SubGrpcsubg 18739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-0g 17142 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-grp 18570 df-minusg 18571 df-subg 18742 |
This theorem is referenced by: (None) |
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