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Mirrors > Home > MPE Home > Th. List > dprdf | Structured version Visualization version GIF version |
Description: The function ๐ is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdf | โข (๐บdom DProd ๐ โ ๐:dom ๐โถ(SubGrpโ๐บ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldmdprd 19695 | . . . . . 6 โข Rel dom DProd | |
2 | 1 | brrelex2i 5675 | . . . . 5 โข (๐บdom DProd ๐ โ ๐ โ V) |
3 | 2 | dmexd 7820 | . . . 4 โข (๐บdom DProd ๐ โ dom ๐ โ V) |
4 | eqid 2736 | . . . 4 โข dom ๐ = dom ๐ | |
5 | eqid 2736 | . . . . 5 โข (Cntzโ๐บ) = (Cntzโ๐บ) | |
6 | eqid 2736 | . . . . 5 โข (0gโ๐บ) = (0gโ๐บ) | |
7 | eqid 2736 | . . . . 5 โข (mrClsโ(SubGrpโ๐บ)) = (mrClsโ(SubGrpโ๐บ)) | |
8 | 5, 6, 7 | dmdprd 19696 | . . . 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 | simp2d 1142 | 1 โข (๐บdom DProd ๐ โ ๐:dom ๐โถ(SubGrpโ๐บ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โง wa 396 โง w3a 1086 = wceq 1540 โ wcel 2105 โwral 3061 Vcvv 3441 โ cdif 3895 โฉ cin 3897 โ wss 3898 {csn 4573 โช cuni 4852 class class class wbr 5092 dom cdm 5620 โ cima 5623 โถwf 6475 โcfv 6479 0gc0g 17247 mrClscmrc 17389 Grpcgrp 18673 SubGrpcsubg 18845 Cntzccntz 19017 DProd cdprd 19691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-ixp 8757 df-dprd 19693 |
This theorem is referenced by: dprdf2 19705 dprdsubg 19722 dprdspan 19725 subgdprd 19733 ablfaclem2 19784 ablfac2 19787 |
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