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Mirrors > Home > MPE Home > Th. List > eldprd | Structured version Visualization version Unicode version |
Description: A class is an internal direct product iff it is the (group) sum of an infinite, but finitely supported cartesian product of subgroups (which mutually commute and have trivial intersections). (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) |
Ref | Expression |
---|---|
dprdval.0 | |
dprdval.w | finSupp |
Ref | Expression |
---|---|
eldprd | DProd DProd g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6220 | . . . . 5 DProd DProd | |
2 | df-ov 6653 | . . . . 5 DProd DProd | |
3 | 1, 2 | eleq2s 2719 | . . . 4 DProd DProd |
4 | df-br 4654 | . . . 4 DProd DProd | |
5 | 3, 4 | sylibr 224 | . . 3 DProd DProd |
6 | 5 | pm4.71ri 665 | . 2 DProd DProd DProd |
7 | dprdval.0 | . . . . . . 7 | |
8 | dprdval.w | . . . . . . 7 finSupp | |
9 | 7, 8 | dprdval 18402 | . . . . . 6 DProd DProd g |
10 | 9 | eleq2d 2687 | . . . . 5 DProd DProd g |
11 | eqid 2622 | . . . . . 6 g g | |
12 | ovex 6678 | . . . . . 6 g | |
13 | 11, 12 | elrnmpti 5376 | . . . . 5 g g |
14 | 10, 13 | syl6bb 276 | . . . 4 DProd DProd g |
15 | 14 | ancoms 469 | . . 3 DProd DProd g |
16 | 15 | pm5.32da 673 | . 2 DProd DProd DProd g |
17 | 6, 16 | syl5bb 272 | 1 DProd DProd g |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 crab 2916 cop 4183 class class class wbr 4653 cmpt 4729 cdm 5114 crn 5115 cfv 5888 (class class class)co 6650 cixp 7908 finSupp cfsupp 8275 c0g 16100 g cgsu 16101 DProd cdprd 18392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-ixp 7909 df-dprd 18394 |
This theorem is referenced by: dprdssv 18415 eldprdi 18417 dprdsubg 18423 dprdss 18428 dmdprdsplitlem 18436 dprddisj2 18438 dpjidcl 18457 |
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