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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 19113 | . . . . . 6 ⊢ Rel dom DProd | |
2 | 1 | brrelex2i 5603 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
3 | 2 | dmexd 7609 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
4 | eqid 2821 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
5 | eqid 2821 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
6 | eqid 2821 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | eqid 2821 | . . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
8 | 5, 6, 7 | dmdprd 19114 | . . . 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 588 | . . 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 269 | . 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 1138 | 1 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 {csn 4560 ∪ cuni 4831 class class class wbr 5058 dom cdm 5549 “ cima 5552 ⟶wf 6345 ‘cfv 6349 0gc0g 16707 mrClscmrc 16848 Grpcgrp 18097 SubGrpcsubg 18267 Cntzccntz 18439 DProd cdprd 19109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-ixp 8456 df-dprd 19111 |
This theorem is referenced by: dprdssv 19132 dprdfid 19133 dprdfinv 19135 dprdfadd 19136 dprdfsub 19137 dprdfeq0 19138 dprdf11 19139 dprdsubg 19140 dprdlub 19142 dprdspan 19143 dprdres 19144 dprdss 19145 dprdf1o 19148 dmdprdsplitlem 19153 dprdcntz2 19154 dprddisj2 19155 dprd2dlem1 19157 dprd2da 19158 dmdprdsplit2lem 19161 dmdprdsplit2 19162 dpjfval 19171 dpjidcl 19174 |
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