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Theorem ablfac 19606

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.)

**Hypotheses**
Ref	Expression
ablfac.b	⊢ 𝐵 = (Base‘𝐺)
ablfac.c	⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
ablfac.1	⊢ (𝜑 → 𝐺 ∈ Abel)
ablfac.2	⊢ (𝜑 → 𝐵 ∈ Fin)

**Assertion**
Ref	Expression
ablfac	⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Distinct variable groups: 𝑠,𝑟,𝐵 𝐶,𝑠 𝜑,𝑠 𝐺,𝑟,𝑠 Allowed substitution hints: 𝜑(𝑟) 𝐶(𝑟)

Proof of Theorem ablfac Dummy variables 𝑝 𝑥 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablfac.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablgrp 19306	. . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		ablfac.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 18672	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
5		ablfac.c	. . . . 5 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
6		ablfac.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin)
7		eqid 2738	. . . . 5 ⊢ (od‘𝐺) = (od‘𝐺)
8		eqid 2738	. . . . 5 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
9		eqid 2738	. . . . 5 ⊢ (𝑝 ∈ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ↦ {𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) = (𝑝 ∈ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ↦ {𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
10		eqid 2738	. . . . 5 ⊢ (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})
11	3, 5, 1, 6, 7, 8, 9, 10	ablfaclem1 19603	. . . 4 ⊢ (𝐵 ∈ (SubGrp‘𝐺) → ((𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})‘𝐵) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)})
12	1, 2, 4, 11	4syl 19	. . 3 ⊢ (𝜑 → ((𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})‘𝐵) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)})
13	3, 5, 1, 6, 7, 8, 9, 10	ablfaclem3 19605	. . 3 ⊢ (𝜑 → ((𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})‘𝐵) ≠ ∅)
14	12, 13	eqnetrrd 3011	. 2 ⊢ (𝜑 → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)} ≠ ∅)
15		rabn0 4316	. 2 ⊢ ({𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)} ≠ ∅ ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
16	14, 15	sylib 217	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 {crab 3067 ∩ cin 3882 ∅c0 4253 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ↑cexp 13710 ♯chash 13972 Word cword 14145 ∥ cdvds 15891 ℙcprime 16304 pCnt cpc 16465 Basecbs 16840 ↾_s cress 16867 Grpcgrp 18492 SubGrpcsubg 18664 odcod 19047 pGrp cpgp 19049 Abelcabl 19302 CycGrpccyg 19392 DProd cdprd 19511

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-rpss 7554 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-0g 17069 df-gsum 17070 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-eqg 18669 df-ghm 18747 df-gim 18790 df-ga 18811 df-cntz 18838 df-oppg 18865 df-od 19051 df-gex 19052 df-pgp 19053 df-lsm 19156 df-pj1 19157 df-cmn 19303 df-abl 19304 df-cyg 19393 df-dprd 19513

This theorem is referenced by: ablfac2 19607

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