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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 19048 | . . . . . 6 ⊢ Rel dom DProd | |
2 | 1 | brrelex2i 5602 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
3 | 2 | dmexd 7604 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
4 | eqid 2818 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
5 | eqid 2818 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
6 | eqid 2818 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | eqid 2818 | . . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
8 | 5, 6, 7 | dmdprd 19049 | . . . 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 268 | . 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 1134 | 1 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 {csn 4557 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 “ cima 5551 ⟶wf 6344 ‘cfv 6348 0gc0g 16701 mrClscmrc 16842 Grpcgrp 18041 SubGrpcsubg 18211 Cntzccntz 18383 DProd cdprd 19044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-ixp 8450 df-dprd 19046 |
This theorem is referenced by: dprdssv 19067 dprdfid 19068 dprdfinv 19070 dprdfadd 19071 dprdfsub 19072 dprdfeq0 19073 dprdf11 19074 dprdsubg 19075 dprdlub 19077 dprdspan 19078 dprdres 19079 dprdss 19080 dprdf1o 19083 dmdprdsplitlem 19088 dprdcntz2 19089 dprddisj2 19090 dprd2dlem1 19092 dprd2da 19093 dmdprdsplit2lem 19096 dmdprdsplit2 19097 dpjfval 19106 dpjidcl 19109 |
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