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Theorem ablfac1lem 19983

Description: Lemma for ablfac1b 19985. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)

**Hypotheses**
Ref	Expression
ablfac1.b	⊢ 𝐵 = (Base‘𝐺)
ablfac1.o	⊢ 𝑂 = (od‘𝐺)
ablfac1.s	⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
ablfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablfac1.f	⊢ (𝜑 → 𝐵 ∈ Fin)
ablfac1.1	⊢ (𝜑 → 𝐴 ⊆ ℙ)
ablfac1.m	⊢ 𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵)))
ablfac1.n	⊢ 𝑁 = ((♯‘𝐵) / 𝑀)

**Assertion**
Ref	Expression
ablfac1lem	⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁)))

Distinct variable groups: 𝑥,𝑝,𝐵 𝜑,𝑝,𝑥 𝐴,𝑝,𝑥 𝑂,𝑝,𝑥 𝑃,𝑝,𝑥 𝐺,𝑝,𝑥 Allowed substitution hints: 𝑆(𝑥,𝑝) 𝑀(𝑥,𝑝) 𝑁(𝑥,𝑝)

**Proof of Theorem ablfac1lem**
Step	Hyp	Ref	Expression
1		ablfac1.m	. . . 4 ⊢ 𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵)))
2		ablfac1.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
3	2	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ)
4		prmnn 16585	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
5	3, 4	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℕ)
6		ablfac1.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19698	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 18879	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac1.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 14273	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
14	10, 13	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
16	3, 15	pccld 16762	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀)
17	5, 16	nnexpcld 14152	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ)
18	1, 17	eqeltrid 2835	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ)
19		ablfac1.n	. . . 4 ⊢ 𝑁 = ((♯‘𝐵) / 𝑀)
20		pcdvds 16776	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
21	3, 15, 20	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∥ (♯‘𝐵))
22	1, 21	eqbrtrid 5126	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (♯‘𝐵))
23		nndivdvds 16172	. . . . . 6 ⊢ (((♯‘𝐵) ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝑀) ∈ ℕ))
24	15, 18, 23	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝑀) ∈ ℕ))
25	22, 24	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((♯‘𝐵) / 𝑀) ∈ ℕ)
26	19, 25	eqeltrid 2835	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ)
27	18, 26	jca 511	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
28	1	oveq1i 7356	. . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁)
29		pcndvds2 16780	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝐵) ∈ ℕ) → ¬ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
30	3, 15, 29	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
31	1	oveq2i 7357	. . . . . . . 8 ⊢ ((♯‘𝐵) / 𝑀) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))
32	19, 31	eqtri 2754	. . . . . . 7 ⊢ 𝑁 = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))
33	32	breq2i 5099	. . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
34	30, 33	sylnibr 329	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁)
35	26	nnzd 12495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ)
36		coprm 16622	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
37	3, 35, 36	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
38	34, 37	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 gcd 𝑁) = 1)
39		prmz 16586	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
40	3, 39	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ)
41		rpexp1i 16634	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀) → ((𝑃 gcd 𝑁) = 1 → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) = 1))
42	40, 35, 16, 41	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃 gcd 𝑁) = 1 → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) = 1))
43	38, 42	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁) = 1)
44	28, 43	eqtrid 2778	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1)
45	19	oveq2i 7357	. . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((♯‘𝐵) / 𝑀))
46	15	nncnd 12141	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℂ)
47	18	nncnd 12141	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ)
48	18	nnne0d 12175	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0)
49	46, 47, 48	divcan2d 11899	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((♯‘𝐵) / 𝑀)) = (♯‘𝐵))
50	45, 49	eqtr2id 2779	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) = (𝑀 · 𝑁))
51	27, 44, 50	3jca 1128	1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 {crab 3395 ⊆ wss 3902 ∅c0 4283 class class class wbr 5091 ↦ cmpt 5172 ‘cfv 6481 (class class class)co 7346 Fincfn 8869 1c1 11007 · cmul 11011 / cdiv 11774 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ↑cexp 13968 ♯chash 14237 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582 pCnt cpc 16748 Basecbs 17120 Grpcgrp 18846 odcod 19437 Abelcabl 19694

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-fz 13408 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-abl 19696

This theorem is referenced by: ablfac1a 19984 ablfac1b 19985

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