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Mirrors > Home > MPE Home > Th. List > dprdf2 | Structured version Visualization version Unicode version |
Description: The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdcntz.1 | DProd |
dprdcntz.2 |
Ref | Expression |
---|---|
dprdf2 | SubGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdcntz.1 | . . 3 DProd | |
2 | dprdf 18405 | . . 3 DProd SubGrp | |
3 | 1, 2 | syl 17 | . 2 SubGrp |
4 | dprdcntz.2 | . . 3 | |
5 | 4 | feq2d 6031 | . 2 SubGrp SubGrp |
6 | 3, 5 | mpbid 222 | 1 SubGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 class class class wbr 4653 cdm 5114 wf 5884 cfv 5888 SubGrpcsubg 17588 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-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-ixp 7909 df-dprd 18394 |
This theorem is referenced by: dprdff 18411 dprdfid 18416 dprdfinv 18418 dprdfadd 18419 dprdfeq0 18421 dprdres 18427 dprdss 18428 dprdf1o 18431 dprdf1 18432 subgdprd 18434 dmdprdsplitlem 18436 dprdcntz2 18437 dpjlem 18450 dpjcntz 18451 dpjdisj 18452 dpjlsm 18453 dpjf 18456 dpjidcl 18457 dpjlid 18460 dpjghm 18462 dpjghm2 18463 ablfac1c 18470 ablfac1eulem 18471 ablfac1eu 18472 ablfaclem2 18485 ablfaclem3 18486 dchrptlem3 24991 |
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