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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 18596 | . . . . . 6 ⊢ Rel dom DProd | |
2 | 1 | brrelex2i 5316 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
3 | dmexg 7262 | . . . . 5 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
5 | eqid 2760 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
6 | eqid 2760 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
7 | eqid 2760 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | eqid 2760 | . . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
9 | 6, 7, 8 | dmdprd 18597 | . . . 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)})))) |
10 | 4, 5, 9 | sylancl 697 | . . 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)})))) |
11 | 10 | ibi 256 | . 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)}))) |
12 | 11 | simp1d 1137 | 1 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∀wral 3050 Vcvv 3340 ∖ cdif 3712 ∩ cin 3714 ⊆ wss 3715 {csn 4321 ∪ cuni 4588 class class class wbr 4804 dom cdm 5266 “ cima 5269 ⟶wf 6045 ‘cfv 6049 0gc0g 16302 mrClscmrc 16445 Grpcgrp 17623 SubGrpcsubg 17789 Cntzccntz 17948 DProd cdprd 18592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-ixp 8075 df-dprd 18594 |
This theorem is referenced by: dprdssv 18615 dprdfid 18616 dprdfinv 18618 dprdfadd 18619 dprdfsub 18620 dprdfeq0 18621 dprdf11 18622 dprdsubg 18623 dprdlub 18625 dprdspan 18626 dprdres 18627 dprdss 18628 dprdf1o 18631 dmdprdsplitlem 18636 dprdcntz2 18637 dprddisj2 18638 dprd2dlem1 18640 dprd2da 18641 dmdprdsplit2lem 18644 dmdprdsplit2 18645 dpjfval 18654 dpjidcl 18657 |
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