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Mirrors > Home > MPE Home > Th. List > odcl2 | Structured version Unicode version |
Description: The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
odcl2.1 |
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odcl2.2 |
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Ref | Expression |
---|---|
odcl2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odcl2.1 |
. . . . . . . . 9
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2 | odcl2.2 |
. . . . . . . . 9
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3 | 1, 2 | odcl 16140 |
. . . . . . . 8
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4 | 3 | adantl 466 |
. . . . . . 7
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5 | elnn0 10679 |
. . . . . . 7
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6 | 4, 5 | sylib 196 |
. . . . . 6
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7 | 6 | ord 377 |
. . . . 5
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8 | eqid 2451 |
. . . . . . . 8
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9 | eqid 2451 |
. . . . . . . 8
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10 | 1, 2, 8, 9 | odinf 16165 |
. . . . . . 7
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11 | 1, 2, 8, 9 | odf1 16164 |
. . . . . . . . 9
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12 | 11 | biimp3a 1319 |
. . . . . . . 8
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13 | f1f 5701 |
. . . . . . . 8
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14 | frn 5660 |
. . . . . . . 8
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15 | ssfi 7631 |
. . . . . . . . 9
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16 | 15 | expcom 435 |
. . . . . . . 8
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17 | 12, 13, 14, 16 | 4syl 21 |
. . . . . . 7
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18 | 10, 17 | mtod 177 |
. . . . . 6
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19 | 18 | 3expia 1190 |
. . . . 5
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20 | 7, 19 | syld 44 |
. . . 4
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21 | 20 | con4d 105 |
. . 3
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22 | 21 | 3impia 1185 |
. 2
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23 | 22 | 3com23 1194 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4498 ax-sep 4508 ax-nul 4516 ax-pow 4565 ax-pr 4626 ax-un 6469 ax-inf2 7945 ax-cnex 9436 ax-resscn 9437 ax-1cn 9438 ax-icn 9439 ax-addcl 9440 ax-addrcl 9441 ax-mulcl 9442 ax-mulrcl 9443 ax-mulcom 9444 ax-addass 9445 ax-mulass 9446 ax-distr 9447 ax-i2m1 9448 ax-1ne0 9449 ax-1rid 9450 ax-rnegex 9451 ax-rrecex 9452 ax-cnre 9453 ax-pre-lttri 9454 ax-pre-lttrn 9455 ax-pre-ltadd 9456 ax-pre-mulgt0 9457 ax-pre-sup 9458 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2599 df-ne 2644 df-nel 2645 df-ral 2798 df-rex 2799 df-reu 2800 df-rmo 2801 df-rab 2802 df-v 3067 df-sbc 3282 df-csb 3384 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-pss 3439 df-nul 3733 df-if 3887 df-pw 3957 df-sn 3973 df-pr 3975 df-tp 3977 df-op 3979 df-uni 4187 df-int 4224 df-iun 4268 df-br 4388 df-opab 4446 df-mpt 4447 df-tr 4481 df-eprel 4727 df-id 4731 df-po 4736 df-so 4737 df-fr 4774 df-se 4775 df-we 4776 df-ord 4817 df-on 4818 df-lim 4819 df-suc 4820 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fn 5516 df-f 5517 df-f1 5518 df-fo 5519 df-f1o 5520 df-fv 5521 df-isom 5522 df-riota 6148 df-ov 6190 df-oprab 6191 df-mpt2 6192 df-om 6574 df-1st 6674 df-2nd 6675 df-recs 6929 df-rdg 6963 df-1o 7017 df-oadd 7021 df-omul 7022 df-er 7198 df-map 7313 df-en 7408 df-dom 7409 df-sdom 7410 df-fin 7411 df-sup 7789 df-oi 7822 df-card 8207 df-acn 8210 df-pnf 9518 df-mnf 9519 df-xr 9520 df-ltxr 9521 df-le 9522 df-sub 9695 df-neg 9696 df-div 10092 df-nn 10421 df-2 10478 df-3 10479 df-n0 10678 df-z 10745 df-uz 10960 df-rp 11090 df-fz 11536 df-fl 11740 df-mod 11807 df-seq 11905 df-exp 11964 df-cj 12687 df-re 12688 df-im 12689 df-sqr 12823 df-abs 12824 df-dvds 13635 df-0g 14479 df-mnd 15514 df-grp 15644 df-minusg 15645 df-sbg 15646 df-mulg 15647 df-od 16133 |
This theorem is referenced by: gexcl2 16189 pgpfi1 16195 odcau 16204 prmcyg 16471 lt6abl 16472 dchrptlem1 22716 dchrptlem2 22717 |
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