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| Mirrors > Home > MPE Home > Th. List > dprdgrp | Structured version Visualization version GIF version | ||
| Description: Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) |
| Ref | Expression |
|---|---|
| dprdgrp | ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reldmdprd 20017 | . . . . . 6 ⊢ Rel dom DProd | |
| 2 | 1 | brrelex2i 5742 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
| 3 | 2 | dmexd 7925 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
| 4 | eqid 2737 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
| 5 | eqid 2737 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 7 | eqid 2737 | . . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
| 8 | 5, 6, 7 | dmdprd 20018 | . . . 4 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)})))) |
| 9 | 3, 4, 8 | sylancl 586 | . . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)})))) |
| 10 | 9 | ibi 267 | . 2 ⊢ (𝐺dom DProd 𝑆 → (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)}))) |
| 11 | 10 | simp1d 1143 | 1 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 “ cima 5688 ⟶wf 6557 ‘cfv 6561 0gc0g 17484 mrClscmrc 17626 Grpcgrp 18951 SubGrpcsubg 19138 Cntzccntz 19333 DProd cdprd 20013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-ixp 8938 df-dprd 20015 |
| This theorem is referenced by: dprdssv 20036 dprdfid 20037 dprdfinv 20039 dprdfadd 20040 dprdfsub 20041 dprdfeq0 20042 dprdf11 20043 dprdsubg 20044 dprdlub 20046 dprdspan 20047 dprdres 20048 dprdss 20049 dprdf1o 20052 dmdprdsplitlem 20057 dprdcntz2 20058 dprddisj2 20059 dprd2dlem1 20061 dprd2da 20062 dmdprdsplit2lem 20065 dmdprdsplit2 20066 dpjfval 20075 dpjidcl 20078 |
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