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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 19112 | . . . . . 6 ⊢ Rel dom DProd | |
2 | 1 | brrelex2i 5573 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
3 | 2 | dmexd 7596 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
4 | eqid 2798 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
5 | eqid 2798 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
6 | eqid 2798 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | eqid 2798 | . . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | |
8 | 5, 6, 7 | dmdprd 19113 | . . . 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 589 | . . 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 270 | . 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 1139 | 1 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 {csn 4525 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 “ cima 5522 ⟶wf 6320 ‘cfv 6324 0gc0g 16705 mrClscmrc 16846 Grpcgrp 18095 SubGrpcsubg 18265 Cntzccntz 18437 DProd cdprd 19108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-ixp 8445 df-dprd 19110 |
This theorem is referenced by: dprdssv 19131 dprdfid 19132 dprdfinv 19134 dprdfadd 19135 dprdfsub 19136 dprdfeq0 19137 dprdf11 19138 dprdsubg 19139 dprdlub 19141 dprdspan 19142 dprdres 19143 dprdss 19144 dprdf1o 19147 dmdprdsplitlem 19152 dprdcntz2 19153 dprddisj2 19154 dprd2dlem1 19156 dprd2da 19157 dmdprdsplit2lem 19160 dmdprdsplit2 19161 dpjfval 19170 dpjidcl 19173 |
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