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Theorem ablfac1lem 20007

Description: Lemma for ablfac1b 20009. 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 3949	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ)
4		prmnn 16651	. . . . . 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 19722	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 18905	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac1.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 14338	. . . . . . . . 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 16828	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (♯‘𝐵)) ∈ ℕ₀)
17	5, 16	nnexpcld 14217	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (♯‘𝐵))) ∈ ℕ)
18	1, 17	eqeltrid 2833	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ)
19		ablfac1.n	. . . 4 ⊢ 𝑁 = ((♯‘𝐵) / 𝑀)
20		pcdvds 16842	. . . . . . 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 5145	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (♯‘𝐵))
23		nndivdvds 16238	. . . . . 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 2833	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ)
27	18, 26	jca 511	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
28	1	oveq1i 7400	. . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (♯‘𝐵))) gcd 𝑁)
29		pcndvds2 16846	. . . . . . 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 7401	. . . . . . . 8 ⊢ ((♯‘𝐵) / 𝑀) = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))
32	19, 31	eqtri 2753	. . . . . . 7 ⊢ 𝑁 = ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵))))
33	32	breq2i 5118	. . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((♯‘𝐵) / (𝑃↑(𝑃 pCnt (♯‘𝐵)))))
34	30, 33	sylnibr 329	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁)
35	26	nnzd 12563	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ)
36		coprm 16688	. . . . . 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 16652	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
40	3, 39	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ)
41		rpexp1i 16700	. . . . 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 2777	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1)
45	19	oveq2i 7401	. . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((♯‘𝐵) / 𝑀))
46	15	nncnd 12209	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘𝐵) ∈ ℂ)
47	18	nncnd 12209	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ)
48	18	nnne0d 12243	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0)
49	46, 47, 48	divcan2d 11967	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((♯‘𝐵) / 𝑀)) = (♯‘𝐵))
50	45, 49	eqtr2id 2778	. 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 1540 ∈ wcel 2109 ≠ wne 2926 {crab 3408 ⊆ wss 3917 ∅c0 4299 class class class wbr 5110 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 1c1 11076 · cmul 11080 / cdiv 11842 ℕcn 12193 ℕ₀cn0 12449 ℤcz 12536 ↑cexp 14033 ♯chash 14302 ∥ cdvds 16229 gcd cgcd 16471 ℙcprime 16648 pCnt cpc 16814 Basecbs 17186 Grpcgrp 18872 odcod 19461 Abelcabl 19718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-fz 13476 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472 df-prm 16649 df-pc 16815 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-abl 19720

This theorem is referenced by: ablfac1a 20008 ablfac1b 20009

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