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Mirrors > Home > MPE Home > Th. List > issubgrpd2 | Structured version Visualization version GIF version |
Description: Prove a subgroup by closure (definition version). (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 |
---|---|
issubgrpd2 | ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubgrpd.ss | . 2 ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) | |
2 | issubgrpd.zcl | . . 3 ⊢ (𝜑 → 0 ∈ 𝐷) | |
3 | 2 | ne0d 4303 | . 2 ⊢ (𝜑 → 𝐷 ≠ ∅) |
4 | issubgrpd.p | . . . . . . . 8 ⊢ (𝜑 → + = (+g‘𝐼)) | |
5 | 4 | oveqd 7175 | . . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐼)𝑦)) |
6 | 5 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) = (𝑥(+g‘𝐼)𝑦)) |
7 | issubgrpd.acl | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) | |
8 | 7 | 3expa 1114 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) |
9 | 6, 8 | eqeltrrd 2916 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥(+g‘𝐼)𝑦) ∈ 𝐷) |
10 | 9 | ralrimiva 3184 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∀𝑦 ∈ 𝐷 (𝑥(+g‘𝐼)𝑦) ∈ 𝐷) |
11 | issubgrpd.ncl | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) | |
12 | 10, 11 | jca 514 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∀𝑦 ∈ 𝐷 (𝑥(+g‘𝐼)𝑦) ∈ 𝐷 ∧ ((invg‘𝐼)‘𝑥) ∈ 𝐷)) |
13 | 12 | ralrimiva 3184 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (∀𝑦 ∈ 𝐷 (𝑥(+g‘𝐼)𝑦) ∈ 𝐷 ∧ ((invg‘𝐼)‘𝑥) ∈ 𝐷)) |
14 | issubgrpd.g | . . 3 ⊢ (𝜑 → 𝐼 ∈ Grp) | |
15 | eqid 2823 | . . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
16 | eqid 2823 | . . . 4 ⊢ (+g‘𝐼) = (+g‘𝐼) | |
17 | eqid 2823 | . . . 4 ⊢ (invg‘𝐼) = (invg‘𝐼) | |
18 | 15, 16, 17 | issubg2 18296 | . . 3 ⊢ (𝐼 ∈ Grp → (𝐷 ∈ (SubGrp‘𝐼) ↔ (𝐷 ⊆ (Base‘𝐼) ∧ 𝐷 ≠ ∅ ∧ ∀𝑥 ∈ 𝐷 (∀𝑦 ∈ 𝐷 (𝑥(+g‘𝐼)𝑦) ∈ 𝐷 ∧ ((invg‘𝐼)‘𝑥) ∈ 𝐷)))) |
19 | 14, 18 | syl 17 | . 2 ⊢ (𝜑 → (𝐷 ∈ (SubGrp‘𝐼) ↔ (𝐷 ⊆ (Base‘𝐼) ∧ 𝐷 ≠ ∅ ∧ ∀𝑥 ∈ 𝐷 (∀𝑦 ∈ 𝐷 (𝑥(+g‘𝐼)𝑦) ∈ 𝐷 ∧ ((invg‘𝐼)‘𝑥) ∈ 𝐷)))) |
20 | 1, 3, 13, 19 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⊆ wss 3938 ∅c0 4293 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 +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: issubgrpd 18298 symgsssg 18597 symgfisg 18598 issubrngd2 19963 dsmmsubg 20889 |
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