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Mirrors > Home > MPE Home > Th. List > sylow3 | Structured version Visualization version Unicode version |
Description: Sylow's third theorem.
The number of Sylow subgroups is a divisor of
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Ref | Expression |
---|---|
sylow3.x |
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sylow3.g |
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sylow3.xf |
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sylow3.p |
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sylow3.n |
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Ref | Expression |
---|---|
sylow3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylow3.g |
. . . 4
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2 | sylow3.xf |
. . . 4
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3 | sylow3.p |
. . . 4
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4 | sylow3.x |
. . . . 5
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5 | 4 | slwn0 17345 |
. . . 4
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6 | 1, 2, 3, 5 | syl3anc 1292 |
. . 3
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7 | n0 3732 |
. . 3
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8 | 6, 7 | sylib 201 |
. 2
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9 | sylow3.n |
. . . 4
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10 | 1 | adantr 472 |
. . . . 5
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11 | 2 | adantr 472 |
. . . . 5
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12 | 3 | adantr 472 |
. . . . 5
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13 | eqid 2471 |
. . . . 5
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14 | eqid 2471 |
. . . . 5
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15 | oveq2 6316 |
. . . . . . . . . 10
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16 | 15 | oveq1d 6323 |
. . . . . . . . 9
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17 | 16 | cbvmptv 4488 |
. . . . . . . 8
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18 | oveq1 6315 |
. . . . . . . . . 10
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19 | id 22 |
. . . . . . . . . 10
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20 | 18, 19 | oveq12d 6326 |
. . . . . . . . 9
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21 | 20 | mpteq2dv 4483 |
. . . . . . . 8
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22 | 17, 21 | syl5eq 2517 |
. . . . . . 7
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23 | 22 | rneqd 5068 |
. . . . . 6
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24 | mpteq1 4476 |
. . . . . . 7
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25 | 24 | rneqd 5068 |
. . . . . 6
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26 | 23, 25 | cbvmpt2v 6390 |
. . . . 5
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27 | simpr 468 |
. . . . 5
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28 | eqid 2471 |
. . . . 5
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29 | eqid 2471 |
. . . . 5
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30 | 4, 10, 11, 12, 13, 14, 26, 27, 28, 29 | sylow3lem4 17360 |
. . . 4
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31 | 9, 30 | syl5eqbr 4429 |
. . 3
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32 | 9 | oveq1i 6318 |
. . . 4
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33 | 23, 25 | cbvmpt2v 6390 |
. . . . 5
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34 | eqid 2471 |
. . . . 5
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35 | 4, 10, 11, 12, 13, 14, 27, 33, 34 | sylow3lem6 17362 |
. . . 4
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36 | 32, 35 | syl5eq 2517 |
. . 3
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37 | 31, 36 | jca 541 |
. 2
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38 | 8, 37 | exlimddv 1789 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-inf2 8164 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 ax-pre-sup 9635 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-fal 1458 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-disj 4367 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-se 4799 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-isom 5598 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-1st 6812 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-1o 7200 df-2o 7201 df-oadd 7204 df-omul 7205 df-er 7381 df-ec 7383 df-qs 7387 df-map 7492 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-sup 7974 df-inf 7975 df-oi 8043 df-card 8391 df-acn 8394 df-cda 8616 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-div 10292 df-nn 10632 df-2 10690 df-3 10691 df-n0 10894 df-z 10962 df-uz 11183 df-q 11288 df-rp 11326 df-fz 11811 df-fzo 11943 df-fl 12061 df-mod 12130 df-seq 12252 df-exp 12311 df-fac 12498 df-bc 12526 df-hash 12554 df-cj 13239 df-re 13240 df-im 13241 df-sqrt 13375 df-abs 13376 df-clim 13629 df-sum 13830 df-dvds 14383 df-gcd 14548 df-prm 14702 df-pc 14866 df-ndx 15202 df-slot 15203 df-base 15204 df-sets 15205 df-ress 15206 df-plusg 15281 df-0g 15418 df-mgm 16566 df-sgrp 16605 df-mnd 16615 df-submnd 16661 df-grp 16751 df-minusg 16752 df-sbg 16753 df-mulg 16754 df-subg 16892 df-nsg 16893 df-eqg 16894 df-ghm 16959 df-ga 17022 df-od 17250 df-pgp 17254 df-slw 17256 |
This theorem is referenced by: (None) |
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