Theorem prmdvdsfmtnof1lem2 47622

Description: Lemma 2 for prmdvdsfmtnof1 47624. (Contributed by AV, 3-Aug-2021.)

Assertion

Ref	Expression
prmdvdsfmtnof1lem2	⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))

Proof of Theorem prmdvdsfmtnof1lem2

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmtnorn 47571	. 2 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2		fmtnorn 47571	. 2 ⊢ (𝐺 ∈ ran FermatNo ↔ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺)
3		2a1 28	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺)))
4	3	2a1d 26	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺)))))
5		fmtnonn 47568	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
6	5	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (FermatNo‘𝑛) ∈ ℕ)
7	6	adantr 480	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑛) ∈ ℕ)
8		eleq1 2819	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
9	8	ad2antll 729	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
10	7, 9	mpbid 232	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11		fmtnonn 47568	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
12	11	ad2antll 729	. . . . . . . . . . 11 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (FermatNo‘𝑚) ∈ ℕ)
13	12	adantr 480	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑚) ∈ ℕ)
14		eleq1 2819	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
15	14	ad2antrl 728	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
16	13, 15	mpbid 232	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19		fveq2 6822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
20	19	con3i 154	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
21	20	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
22	21	neqned 2935	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ≠ 𝑚)
23		goldbachth 47584	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑛 ≠ 𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
24	17, 18, 22, 23	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
25	24	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26		eqeq12 2748	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
27	26	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28		oveq12 7355	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
29	28	eqeq1d 2733	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
30	27, 29	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
31	30	ancoms 458	. . . . . . . . . . . . 13 ⊢ (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
32	25, 31	syl5ibcom 245	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
33	32	com23 86	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → (¬ 𝐹 = 𝐺 → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1)))
34	33	impcom 407	. . . . . . . . . 10 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
35	34	imp 406	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36		prmnn 16585	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37		coprmdvds1 16563	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐼 = 1))
38	37	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
39	36, 38	syl3anr1 1418	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
40		eleq1 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41		1nprm 16590	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 1 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℙ → 𝐹 = 𝐺)
43	40, 42	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝐼 = 1 → (𝐼 ∈ ℙ → 𝐹 = 𝐺))
44	43	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ ℙ → (𝐼 = 1 → 𝐹 = 𝐺))
45	44	a1d 25	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ ℙ → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
46	45	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
47	46	impcom 407	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → (𝐼 = 1 → 𝐹 = 𝐺))
48	39, 47	mpd 15	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐹 = 𝐺)
49	48	ex 412	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
50	10, 16, 35, 49	syl3anc 1373	. . . . . . . 8 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
51	50	exp43 436	. . . . . . 7 ⊢ (¬ 𝐹 = 𝐺 → ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺)))))
52	4, 51	pm2.61i 182	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))))
53	52	rexlimdva 3133	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))))
54	53	com23 86	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))))
55	54	rexlimiv 3126	. . 3 ⊢ (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺)))
56	55	imp 406	. 2 ⊢ ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
57	1, 2, 56	syl2anb 598	1 ⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   class class class wbr 5091  ran crn 5617  ‘cfv 6481  (class class class)co 7346  1c1 11007  ℕcn 12125  ℕ₀cn0 12381   ∥ cdvds 16163   gcd cgcd 16405  ℙcprime 16582  FermatNocfmtno 47564

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-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  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-se 5570  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-isom 6490  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-oi 9396  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-4 12190  df-5 12191  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-fz 13408  df-fzo 13555  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-clim 15395  df-prod 15811  df-dvds 16164  df-gcd 16406  df-prm 16583  df-fmtno 47565

This theorem is referenced by:  prmdvdsfmtnof1  47624