Theorem prmdvdsfmtnof1lem2 47566

Description: Lemma 2 for prmdvdsfmtnof1 47568. (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 47515	. 2 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2		fmtnorn 47515	. 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 47512	. . . . . . . . . . . 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 2823	. . . . . . . . . . 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 47512	. . . . . . . . . . . 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 2823	. . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . 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 2940	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ≠ 𝑚)
23		goldbachth 47528	. . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
29	28	eqeq1d 2738	. . . . . . . . . . . . . . 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 16698	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37		coprmdvds1 16676	. . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41		1nprm 16703	. . . . . . . . . . . . . . . . 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 3142	. . . . 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 3135	. . 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 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061   class class class wbr 5124  ran crn 5660  ‘cfv 6536  (class class class)co 7410  1c1 11135  ℕcn 12245  ℕ₀cn0 12506   ∥ cdvds 16277   gcd cgcd 16518  ℙcprime 16695  FermatNocfmtno 47508

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14354  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-clim 15509  df-prod 15925  df-dvds 16278  df-gcd 16519  df-prm 16696  df-fmtno 47509

This theorem is referenced by:  prmdvdsfmtnof1  47568