Theorem prmdvdsfmtnof1lem2 44925

Description: Lemma 2 for prmdvdsfmtnof1 44927. (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 44874	. 2 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2		fmtnorn 44874	. 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 44871	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
6	5	ad2antrl 724	. . . . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
9	8	ad2antll 725	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
10	7, 9	mpbid 231	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11		fmtnonn 44871	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
12	11	ad2antll 725	. . . . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
15	14	ad2antrl 724	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
16	13, 15	mpbid 231	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17		simpll 763	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19		fveq2 6756	. . . . . . . . . . . . . . . . . 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 2949	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ≠ 𝑚)
23		goldbachth 44887	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑛 ≠ 𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
24	17, 18, 22, 23	syl3anc 1369	. . . . . . . . . . . . . 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 2755	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
27	26	notbid 317	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28		oveq12 7264	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
29	28	eqeq1d 2740	. . . . . . . . . . . . . . 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 244	. . . . . . . . . . . 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 16307	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37		coprmdvds1 16285	. . . . . . . . . . . . 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 1414	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
40		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41		1nprm 16312	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 1 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℙ → 𝐹 = 𝐺)
43	40, 42	syl6bi 252	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 1369	. . . . . . . 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 3212	. . . . 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 3208	. . 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 597	1 ⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   class class class wbr 5070  ran crn 5581  ‘cfv 6418  (class class class)co 7255  1c1 10803  ℕcn 11903  ℕ₀cn0 12163   ∥ cdvds 15891   gcd cgcd 16129  ℙcprime 16304  FermatNocfmtno 44867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-prod 15544  df-dvds 15892  df-gcd 16130  df-prm 16305  df-fmtno 44868

This theorem is referenced by:  prmdvdsfmtnof1  44927