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Theorem ablfac 20059

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 19754	. . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3		ablfac.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgid 19099	. . . 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 2741	. . . . 5 ⊢ (od‘𝐺) = (od‘𝐺)
8		eqid 2741	. . . . 5 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
9		eqid 2741	. . . . 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 2741	. . . . 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 20056	. . . 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 20058	. . 3 ⊢ (𝜑 → ((𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})‘𝐵) ≠ ∅)
14	12, 13	eqnetrrd 3004	. 2 ⊢ (𝜑 → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)} ≠ ∅)
15		rabn0 4319	. 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 220	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 {crab 3393 ∩ cin 3883 ∅c0 4263 class class class wbr 5074 ↦ cmpt 5155 dom cdm 5620 ran crn 5621 ‘cfv 6488 (class class class)co 7359 Fincfn 8887 ↑cexp 14018 ♯chash 14287 Word cword 14470 ∥ cdvds 16216 ℙcprime 16635 pCnt cpc 16802 Basecbs 17174 ↾_s cress 17195 Grpcgrp 18904 SubGrpcsubg 19091 odcod 19493 pGrp cpgp 19495 Abelcabl 19750 CycGrpccyg 19846 DProd cdprd 19964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-inf2 9557 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-rpss 7669 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-acn 9861 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-q 12894 df-rp 12938 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-word 14471 df-concat 14528 df-s1 14554 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-dvds 16217 df-gcd 16459 df-prm 16636 df-pc 16803 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-0g 17399 df-gsum 17400 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-eqg 19096 df-ghm 19183 df-gim 19228 df-ga 19259 df-cntz 19286 df-oppg 19315 df-od 19497 df-gex 19498 df-pgp 19499 df-lsm 19605 df-pj1 19606 df-cmn 19751 df-abl 19752 df-cyg 19847 df-dprd 19966

This theorem is referenced by: ablfac2 20060

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