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Mirrors > Home > MPE Home > Th. List > ablfac | Structured version Unicode version |
Description: The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
Ref | Expression |
---|---|
ablfac.b |
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ablfac.c |
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ablfac.1 |
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ablfac.2 |
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Ref | Expression |
---|---|
ablfac |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablfac.1 |
. . . 4
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2 | ablgrp 16388 |
. . . 4
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3 | ablfac.b |
. . . . 5
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4 | 3 | subgid 15787 |
. . . 4
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5 | ablfac.c |
. . . . 5
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6 | ablfac.2 |
. . . . 5
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7 | eqid 2451 |
. . . . 5
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8 | eqid 2451 |
. . . . 5
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9 | eqid 2451 |
. . . . 5
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10 | eqid 2451 |
. . . . 5
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11 | 3, 5, 1, 6, 7, 8, 9, 10 | ablfaclem1 16693 |
. . . 4
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12 | 1, 2, 4, 11 | 4syl 21 |
. . 3
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13 | 3, 5, 1, 6, 7, 8, 9, 10 | ablfaclem3 16695 |
. . 3
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14 | 12, 13 | eqnetrrd 2742 |
. 2
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15 | rabn0 3757 |
. 2
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16 | 14, 15 | sylib 196 |
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 4503 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-inf2 7950 ax-cnex 9441 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 ax-pre-sup 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 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 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-pss 3444 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-tp 3982 df-op 3984 df-uni 4192 df-int 4229 df-iun 4273 df-iin 4274 df-disj 4363 df-br 4393 df-opab 4451 df-mpt 4452 df-tr 4486 df-eprel 4732 df-id 4736 df-po 4741 df-so 4742 df-fr 4779 df-se 4780 df-we 4781 df-ord 4822 df-on 4823 df-lim 4824 df-suc 4825 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-isom 5527 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-of 6422 df-rpss 6462 df-om 6579 df-1st 6679 df-2nd 6680 df-supp 6793 df-tpos 6847 df-recs 6934 df-rdg 6968 df-1o 7022 df-2o 7023 df-oadd 7026 df-omul 7027 df-er 7203 df-ec 7205 df-qs 7209 df-map 7318 df-pm 7319 df-ixp 7366 df-en 7413 df-dom 7414 df-sdom 7415 df-fin 7416 df-fsupp 7724 df-sup 7794 df-oi 7827 df-card 8212 df-acn 8215 df-cda 8440 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-div 10097 df-nn 10426 df-2 10483 df-3 10484 df-n0 10683 df-z 10750 df-uz 10965 df-q 11057 df-rp 11095 df-fz 11541 df-fzo 11652 df-fl 11745 df-mod 11812 df-seq 11910 df-exp 11969 df-fac 12155 df-bc 12182 df-hash 12207 df-word 12333 df-concat 12335 df-s1 12336 df-cj 12692 df-re 12693 df-im 12694 df-sqr 12828 df-abs 12829 df-clim 13070 df-sum 13268 df-dvds 13640 df-gcd 13795 df-prm 13868 df-pc 14008 df-ndx 14281 df-slot 14282 df-base 14283 df-sets 14284 df-ress 14285 df-plusg 14355 df-0g 14484 df-gsum 14485 df-mre 14628 df-mrc 14629 df-acs 14631 df-mnd 15519 df-mhm 15568 df-submnd 15569 df-grp 15649 df-minusg 15650 df-sbg 15651 df-mulg 15652 df-subg 15782 df-eqg 15784 df-ghm 15849 df-gim 15891 df-ga 15912 df-cntz 15939 df-oppg 15965 df-od 16138 df-gex 16139 df-pgp 16140 df-lsm 16241 df-pj1 16242 df-cmn 16385 df-abl 16386 df-cyg 16461 df-dprd 16584 |
This theorem is referenced by: ablfac2 16697 |
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