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Theorem ablfac 20060

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 19755	. . . 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 2737	. . . . 5 ⊢ (od‘𝐺) = (od‘𝐺)
8		eqid 2737	. . . . 5 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
9		eqid 2737	. . . . 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 2737	. . . . 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 20057	. . . 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 20059	. . 3 ⊢ (𝜑 → ((𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})‘𝐵) ≠ ∅)
14	12, 13	eqnetrrd 3001	. 2 ⊢ (𝜑 → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)} ≠ ∅)
15		rabn0 4330	. 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 218	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 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 {crab 3390 ∩ cin 3889 ∅c0 4274 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5626 ran crn 5627 ‘cfv 6494 (class class class)co 7362 Fincfn 8888 ↑cexp 14018 ♯chash 14287 Word cword 14470 ∥ cdvds 16216 ℙcprime 16635 pCnt cpc 16802 Basecbs 17174 ↾_s cress 17195 Grpcgrp 18904 SubGrpcsubg 19091 odcod 19494 pGrp cpgp 19496 Abelcabl 19751 CycGrpccyg 19847 DProd cdprd 19965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-rpss 7672 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-omul 8405 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9820 df-card 9858 df-acn 9861 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 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 19229 df-ga 19260 df-cntz 19287 df-oppg 19316 df-od 19498 df-gex 19499 df-pgp 19500 df-lsm 19606 df-pj1 19607 df-cmn 19752 df-abl 19753 df-cyg 19848 df-dprd 19967

This theorem is referenced by: ablfac2 20061

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