Theorem prmdvdsfmtnof1lem2 41497

Description: Lemma 2 for prmdvdsfmtnof1 41499. (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 41446	. 2 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2		fmtnorn 41446	. 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 41443	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
6	5	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (FermatNo‘𝑛) ∈ ℕ)
7	6	adantr 481	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑛) ∈ ℕ)
8		eleq1 2689	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
9	8	ad2antll 765	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
10	7, 9	mpbid 222	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11		fmtnonn 41443	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
12	11	ad2antll 765	. . . . . . . . . . 11 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (FermatNo‘𝑚) ∈ ℕ)
13	12	adantr 481	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑚) ∈ ℕ)
14		eleq1 2689	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
15	14	ad2antrl 764	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
16	13, 15	mpbid 222	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17		simpll 790	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18		simplr 792	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
20	19	con3i 150	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
21	20	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
22	21	neqned 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ≠ 𝑚)
23		goldbachth 41459	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑛 ≠ 𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
24	17, 18, 22, 23	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
25	24	ex 450	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26		eqeq12 2635	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
27	26	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28		oveq12 6659	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
29	28	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
30	27, 29	imbi12d 334	. . . . . . . . . . . . . 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 469	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 446	. . . . . . . . . 10 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
35	34	imp 445	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36		prmnn 15388	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37		coprmdvds1 15365	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐼 = 1))
38	37	imp 445	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
39	36, 38	syl3anr1 1378	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
40		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41		1nprm 15392	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 1 ∈ ℙ
42	41	pm2.21i 116	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℙ → 𝐹 = 𝐺)
43	40, 42	syl6bi 243	. . . . . . . . . . . . . . 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 1082	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
47	46	impcom 446	. . . . . . . . . . 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 450	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
50	10, 16, 35, 49	syl3anc 1326	. . . . . . . 8 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
51	50	exp43 640	. . . . . . 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 176	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))))
53	52	rexlimdva 3031	. . . . 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 3027	. . 3 ⊢ (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺)))
56	55	imp 445	. 2 ⊢ ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
57	1, 2, 56	syl2anb 496	1 ⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   class class class wbr 4653  ran crn 5115  ‘cfv 5888  (class class class)co 6650  1c1 9937  ℕcn 11020  ℕ₀cn0 11292   ∥ cdvds 14983   gcd cgcd 15216  ℙcprime 15385  FermatNocfmtno 41439

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-fal 1489  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-se 5074  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-isom 5897  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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  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-4 11081  df-5 11082  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-prm 15386  df-fmtno 41440

This theorem is referenced by:  prmdvdsfmtnof1  41499