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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 19869 | . . . . . 6 โข Rel dom DProd | |
2 | 1 | brrelex2i 5733 | . . . . 5 โข (๐บdom DProd ๐ โ ๐ โ V) |
3 | 2 | dmexd 7898 | . . . 4 โข (๐บdom DProd ๐ โ dom ๐ โ V) |
4 | eqid 2732 | . . . 4 โข dom ๐ = dom ๐ | |
5 | eqid 2732 | . . . . 5 โข (Cntzโ๐บ) = (Cntzโ๐บ) | |
6 | eqid 2732 | . . . . 5 โข (0gโ๐บ) = (0gโ๐บ) | |
7 | eqid 2732 | . . . . 5 โข (mrClsโ(SubGrpโ๐บ)) = (mrClsโ(SubGrpโ๐บ)) | |
8 | 5, 6, 7 | dmdprd 19870 | . . . 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 586 | . . 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 266 | . 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 1142 | 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 396 โง w3a 1087 = wceq 1541 โ wcel 2106 โwral 3061 Vcvv 3474 โ cdif 3945 โฉ cin 3947 โ wss 3948 {csn 4628 โช cuni 4908 class class class wbr 5148 dom cdm 5676 โ cima 5679 โถwf 6539 โcfv 6543 0gc0g 17387 mrClscmrc 17529 Grpcgrp 18821 SubGrpcsubg 19002 Cntzccntz 19181 DProd cdprd 19865 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-ixp 8894 df-dprd 19867 |
This theorem is referenced by: dprdssv 19888 dprdfid 19889 dprdfinv 19891 dprdfadd 19892 dprdfsub 19893 dprdfeq0 19894 dprdf11 19895 dprdsubg 19896 dprdlub 19898 dprdspan 19899 dprdres 19900 dprdss 19901 dprdf1o 19904 dmdprdsplitlem 19909 dprdcntz2 19910 dprddisj2 19911 dprd2dlem1 19913 dprd2da 19914 dmdprdsplit2lem 19917 dmdprdsplit2 19918 dpjfval 19927 dpjidcl 19930 |
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