ablfac1lem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ablfac1lem		Structured version Visualization version GIF version

Theorem ablfac1lem 18467

Description: Lemma for ablfac1b 18469. 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 3603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℙ)
4		prmnn 15388	. . . . . 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 18198	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 17451	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac1.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 13157	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
14	10, 13	mpbird 247	. . . . . . 7 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘𝐵) ∈ ℕ)
16	3, 15	pccld 15555	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 pCnt (#‘𝐵)) ∈ ℕ₀)
17	5, 16	nnexpcld 13030	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (#‘𝐵))) ∈ ℕ)
18	1, 17	syl5eqel 2705	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℕ)
19		ablfac1.n	. . . 4 ⊢ 𝑁 = ((#‘𝐵) / 𝑀)
20		pcdvds 15568	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → (𝑃↑(𝑃 pCnt (#‘𝐵))) ∥ (#‘𝐵))
21	3, 15, 20	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃↑(𝑃 pCnt (#‘𝐵))) ∥ (#‘𝐵))
22	1, 21	syl5eqbr 4688	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∥ (#‘𝐵))
23		nndivdvds 14989	. . . . . 6 ⊢ (((#‘𝐵) ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ (#‘𝐵) ↔ ((#‘𝐵) / 𝑀) ∈ ℕ))
24	15, 18, 23	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∥ (#‘𝐵) ↔ ((#‘𝐵) / 𝑀) ∈ ℕ))
25	22, 24	mpbid 222	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((#‘𝐵) / 𝑀) ∈ ℕ)
26	19, 25	syl5eqel 2705	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℕ)
27	18, 26	jca 554	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
28	1	oveq1i 6660	. . 3 ⊢ (𝑀 gcd 𝑁) = ((𝑃↑(𝑃 pCnt (#‘𝐵))) gcd 𝑁)
29		pcndvds2 15572	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → ¬ 𝑃 ∥ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))))
30	3, 15, 29	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))))
31	1	oveq2i 6661	. . . . . . . 8 ⊢ ((#‘𝐵) / 𝑀) = ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))
32	19, 31	eqtri 2644	. . . . . . 7 ⊢ 𝑁 = ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵))))
33	32	breq2i 4661	. . . . . 6 ⊢ (𝑃 ∥ 𝑁 ↔ 𝑃 ∥ ((#‘𝐵) / (𝑃↑(𝑃 pCnt (#‘𝐵)))))
34	30, 33	sylnibr 319	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ∥ 𝑁)
35	26	nnzd 11481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑁 ∈ ℤ)
36		coprm 15423	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
37	3, 35, 36	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
38	34, 37	mpbid 222	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑃 gcd 𝑁) = 1)
39		prmz 15389	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
40	3, 39	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ℤ)
41		rpexp1i 15433	. . . . 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 1326	. . . 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	syl5eq 2668	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 gcd 𝑁) = 1)
45	19	oveq2i 6661	. . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · ((#‘𝐵) / 𝑀))
46	15	nncnd 11036	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘𝐵) ∈ ℂ)
47	18	nncnd 11036	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ∈ ℂ)
48	18	nnne0d 11065	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → 𝑀 ≠ 0)
49	46, 47, 48	divcan2d 10803	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (𝑀 · ((#‘𝐵) / 𝑀)) = (#‘𝐵))
50	45, 49	syl5req 2669	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (#‘𝐵) = (𝑀 · 𝑁))
51	27, 44, 50	3jca 1242	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 1c1 9937 · cmul 9941 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 ∥ cdvds 14983 gcd cgcd 15216 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 Grpcgrp 17422 odcod 17944 Abelcabl 18194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-abl 18196

This theorem is referenced by: ablfac1a 18468 ablfac1b 18469

W3C validator