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Mirrors > Home > MPE Home > Th. List > dprdpr | Unicode version |
Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dmdprdpr.z |
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dmdprdpr.0 |
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dmdprdpr.s |
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dmdprdpr.t |
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dprdpr.s |
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dprdpr.1 |
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dprdpr.2 |
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Ref | Expression |
---|---|
dprdpr |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmdprdpr.s |
. . . 4
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2 | dmdprdpr.t |
. . . 4
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3 | xpscf 13746 |
. . . 4
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4 | 1, 2, 3 | sylanbrc 646 |
. . 3
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5 | 1n0 6698 |
. . . . 5
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6 | 5 | necomi 2649 |
. . . 4
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7 | disjsn2 3829 |
. . . 4
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8 | 6, 7 | mp1i 12 |
. . 3
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9 | df2o3 6696 |
. . . . 5
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10 | df-pr 3781 |
. . . . 5
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11 | 9, 10 | eqtri 2424 |
. . . 4
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12 | 11 | a1i 11 |
. . 3
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13 | dprdpr.s |
. . 3
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14 | dprdpr.1 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | dprdpr.2 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | dmdprdpr.z |
. . . . 5
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17 | dmdprdpr.0 |
. . . . 5
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18 | 16, 17, 1, 2 | dmdprdpr 15562 |
. . . 4
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19 | 14, 15, 18 | mpbir2and 889 |
. . 3
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20 | 4, 8, 12, 13, 19 | dprdsplit 15561 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | ffn 5550 |
. . . . . . . 8
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22 | 4, 21 | syl 16 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 0ex 4299 |
. . . . . . . . 9
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24 | 23 | prid1 3872 |
. . . . . . . 8
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25 | 24, 9 | eleqtrri 2477 |
. . . . . . 7
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26 | fnressn 5877 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 22, 25, 26 | sylancl 644 |
. . . . . 6
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28 | xpsc0 13740 |
. . . . . . . . 9
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29 | 1, 28 | syl 16 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | opeq2d 3951 |
. . . . . . 7
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31 | 30 | sneqd 3787 |
. . . . . 6
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32 | 27, 31 | eqtrd 2436 |
. . . . 5
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33 | 32 | oveq2d 6056 |
. . . 4
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34 | dprdsn 15549 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
35 | 23, 1, 34 | sylancr 645 |
. . . . 5
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36 | 35 | simprd 450 |
. . . 4
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37 | 33, 36 | eqtrd 2436 |
. . 3
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38 | 1on 6690 |
. . . . . . . . . 10
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39 | 38 | elexi 2925 |
. . . . . . . . 9
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40 | 39 | prid2 3873 |
. . . . . . . 8
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41 | 40, 9 | eleqtrri 2477 |
. . . . . . 7
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42 | fnressn 5877 |
. . . . . . 7
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43 | 22, 41, 42 | sylancl 644 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | xpsc1 13741 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
45 | 2, 44 | syl 16 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 45 | opeq2d 3951 |
. . . . . . 7
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47 | 46 | sneqd 3787 |
. . . . . 6
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48 | 43, 47 | eqtrd 2436 |
. . . . 5
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49 | 48 | oveq2d 6056 |
. . . 4
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50 | dprdsn 15549 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
51 | 38, 2, 50 | sylancr 645 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 51 | simprd 450 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 49, 52 | eqtrd 2436 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 37, 53 | oveq12d 6058 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 20, 54 | eqtrd 2436 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-inf2 7552 ax-cnex 9002 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-pss 3296 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-tp 3782 df-op 3783 df-uni 3976 df-int 4011 df-iun 4055 df-iin 4056 df-br 4173 df-opab 4227 df-mpt 4228 df-tr 4263 df-eprel 4454 df-id 4458 df-po 4463 df-so 4464 df-fr 4501 df-se 4502 df-we 4503 df-ord 4544 df-on 4545 df-lim 4546 df-suc 4547 df-om 4805 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-isom 5422 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-of 6264 df-1st 6308 df-2nd 6309 df-tpos 6438 df-riota 6508 df-recs 6592 df-rdg 6627 df-1o 6683 df-2o 6684 df-oadd 6687 df-er 6864 df-map 6979 df-ixp 7023 df-en 7069 df-dom 7070 df-sdom 7071 df-fin 7072 df-oi 7435 df-card 7782 df-cda 8004 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 df-nn 9957 df-2 10014 df-n0 10178 df-z 10239 df-uz 10445 df-fz 11000 df-fzo 11091 df-seq 11279 df-hash 11574 df-ndx 13427 df-slot 13428 df-base 13429 df-sets 13430 df-ress 13431 df-plusg 13497 df-0g 13682 df-gsum 13683 df-mre 13766 df-mrc 13767 df-acs 13769 df-mnd 14645 df-mhm 14693 df-submnd 14694 df-grp 14767 df-minusg 14768 df-sbg 14769 df-mulg 14770 df-subg 14896 df-ghm 14959 df-gim 15001 df-cntz 15071 df-oppg 15097 df-lsm 15225 df-cmn 15369 df-dprd 15511 |
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