Theorem prmdvdsfmtnof1lem2 42500

Description: Lemma 2 for prmdvdsfmtnof1 42502. (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 42449	. 2 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2		fmtnorn 42449	. 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 42446	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
6	5	ad2antrl 718	. . . . . . . . . . 11 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (FermatNo‘𝑛) ∈ ℕ)
7	6	adantr 474	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑛) ∈ ℕ)
8		eleq1 2846	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
9	8	ad2antll 719	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
10	7, 9	mpbid 224	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11		fmtnonn 42446	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
12	11	ad2antll 719	. . . . . . . . . . 11 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (FermatNo‘𝑚) ∈ ℕ)
13	12	adantr 474	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑚) ∈ ℕ)
14		eleq1 2846	. . . . . . . . . . 11 ⊢ ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
15	14	ad2antrl 718	. . . . . . . . . 10 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
16	13, 15	mpbid 224	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17		simpll 757	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18		simplr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19		fveq2 6446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
20	19	con3i 152	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
21	20	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
22	21	neqned 2975	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ≠ 𝑚)
23		goldbachth 42462	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑛 ≠ 𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
24	17, 18, 22, 23	syl3anc 1439	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
25	24	ex 403	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26		eqeq12 2790	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
27	26	notbid 310	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28		oveq12 6931	. . . . . . . . . . . . . . . 16 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
29	28	eqeq1d 2779	. . . . . . . . . . . . . . 15 ⊢ (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
30	27, 29	imbi12d 336	. . . . . . . . . . . . . 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 452	. . . . . . . . . . . . 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 237	. . . . . . . . . . . 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 398	. . . . . . . . . 10 ⊢ ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
35	34	imp 397	. . . . . . . . 9 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36		prmnn 15793	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37		coprmdvds1 15771	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐼 = 1))
38	37	imp 397	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
39	36, 38	syl3anr1 1487	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)) → 𝐼 = 1)
40		eleq1 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41		1nprm 15797	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 1 ∈ ℙ
42	41	pm2.21i 117	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℙ → 𝐹 = 𝐺)
43	40, 42	syl6bi 245	. . . . . . . . . . . . . . 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 1124	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
47	46	impcom 398	. . . . . . . . . . 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 403	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
50	10, 16, 35, 49	syl3anc 1439	. . . . . . . 8 ⊢ (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
51	50	exp43 429	. . . . . . 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 177	. . . . . 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 397	. 2 ⊢ ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
57	1, 2, 56	syl2anb 591	1 ⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∃wrex 3090   class class class wbr 4886  ran crn 5356  ‘cfv 6135  (class class class)co 6922  1c1 10273  ℕcn 11374  ℕ₀cn0 11642   ∥ cdvds 15387   gcd cgcd 15622  ℙcprime 15790  FermatNocfmtno 42442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-fz 12644  df-fzo 12785  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-prod 15039  df-dvds 15388  df-gcd 15623  df-prm 15791  df-fmtno 42443

This theorem is referenced by:  prmdvdsfmtnof1  42502