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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 | โข (๐บdom DProd ๐ โ ๐บ โ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldmdprd 19867 | . . . . . 6 โข Rel dom DProd | |
2 | 1 | brrelex2i 5734 | . . . . 5 โข (๐บdom DProd ๐ โ ๐ โ V) |
3 | 2 | dmexd 7896 | . . . 4 โข (๐บdom DProd ๐ โ dom ๐ โ V) |
4 | eqid 2733 | . . . 4 โข dom ๐ = dom ๐ | |
5 | eqid 2733 | . . . . 5 โข (Cntzโ๐บ) = (Cntzโ๐บ) | |
6 | eqid 2733 | . . . . 5 โข (0gโ๐บ) = (0gโ๐บ) | |
7 | eqid 2733 | . . . . 5 โข (mrClsโ(SubGrpโ๐บ)) = (mrClsโ(SubGrpโ๐บ)) | |
8 | 5, 6, 7 | dmdprd 19868 | . . . 4 โข ((dom ๐ โ V โง dom ๐ = dom ๐) โ (๐บdom DProd ๐ โ (๐บ โ Grp โง ๐:dom ๐โถ(SubGrpโ๐บ) โง โ๐ฅ โ dom ๐(โ๐ฆ โ (dom ๐ โ {๐ฅ})(๐โ๐ฅ) โ ((Cntzโ๐บ)โ(๐โ๐ฆ)) โง ((๐โ๐ฅ) โฉ ((mrClsโ(SubGrpโ๐บ))โโช (๐ โ (dom ๐ โ {๐ฅ})))) = {(0gโ๐บ)})))) |
9 | 3, 4, 8 | sylancl 587 | . . 3 โข (๐บdom DProd ๐ โ (๐บdom DProd ๐ โ (๐บ โ Grp โง ๐:dom ๐โถ(SubGrpโ๐บ) โง โ๐ฅ โ dom ๐(โ๐ฆ โ (dom ๐ โ {๐ฅ})(๐โ๐ฅ) โ ((Cntzโ๐บ)โ(๐โ๐ฆ)) โง ((๐โ๐ฅ) โฉ ((mrClsโ(SubGrpโ๐บ))โโช (๐ โ (dom ๐ โ {๐ฅ})))) = {(0gโ๐บ)})))) |
10 | 9 | ibi 267 | . 2 โข (๐บdom DProd ๐ โ (๐บ โ Grp โง ๐:dom ๐โถ(SubGrpโ๐บ) โง โ๐ฅ โ dom ๐(โ๐ฆ โ (dom ๐ โ {๐ฅ})(๐โ๐ฅ) โ ((Cntzโ๐บ)โ(๐โ๐ฆ)) โง ((๐โ๐ฅ) โฉ ((mrClsโ(SubGrpโ๐บ))โโช (๐ โ (dom ๐ โ {๐ฅ})))) = {(0gโ๐บ)}))) |
11 | 10 | simp1d 1143 | 1 โข (๐บdom DProd ๐ โ ๐บ โ Grp) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โง wa 397 โง w3a 1088 = wceq 1542 โ wcel 2107 โwral 3062 Vcvv 3475 โ cdif 3946 โฉ cin 3948 โ wss 3949 {csn 4629 โช cuni 4909 class class class wbr 5149 dom cdm 5677 โ cima 5680 โถwf 6540 โcfv 6544 0gc0g 17385 mrClscmrc 17527 Grpcgrp 18819 SubGrpcsubg 19000 Cntzccntz 19179 DProd cdprd 19863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-ixp 8892 df-dprd 19865 |
This theorem is referenced by: dprdssv 19886 dprdfid 19887 dprdfinv 19889 dprdfadd 19890 dprdfsub 19891 dprdfeq0 19892 dprdf11 19893 dprdsubg 19894 dprdlub 19896 dprdspan 19897 dprdres 19898 dprdss 19899 dprdf1o 19902 dmdprdsplitlem 19907 dprdcntz2 19908 dprddisj2 19909 dprd2dlem1 19911 dprd2da 19912 dmdprdsplit2lem 19915 dmdprdsplit2 19916 dpjfval 19925 dpjidcl 19928 |
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