nfermltl2rev - Mathbox for Alexander van der Vekens

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Theorem nfermltl2rev 47083

Description: Fermat's little theorem with base 2 reversed is not generally true: There is an integer 𝑝 (for example 341, see 341fppr2 47074) so that "𝑝 is prime" does not follow from 2↑𝑝≡2 (mod 𝑝). (Contributed by AV, 3-Jun-2023.)

**Assertion**
Ref	Expression
nfermltl2rev	⊢ ∃𝑝 ∈ (ℤ_≥‘3) ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ)

**Proof of Theorem nfermltl2rev**
Step	Hyp	Ref	Expression
1		decex 12711	. . 3 ⊢ ;;341 ∈ V
2		eleq1 2817	. . . 4 ⊢ (𝑝 = ;;341 → (𝑝 ∈ (ℤ_≥‘3) ↔ ;;341 ∈ (ℤ_≥‘3)))
3		oveq2 7428	. . . . . . . 8 ⊢ (𝑝 = ;;341 → (2↑𝑝) = (2↑;;341))
4		id 22	. . . . . . . 8 ⊢ (𝑝 = ;;341 → 𝑝 = ;;341)
5	3, 4	oveq12d 7438	. . . . . . 7 ⊢ (𝑝 = ;;341 → ((2↑𝑝) mod 𝑝) = ((2↑;;341) mod ;;341))
6		oveq2 7428	. . . . . . 7 ⊢ (𝑝 = ;;341 → (2 mod 𝑝) = (2 mod ;;341))
7	5, 6	eqeq12d 2744	. . . . . 6 ⊢ (𝑝 = ;;341 → (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) ↔ ((2↑;;341) mod ;;341) = (2 mod ;;341)))
8		eleq1 2817	. . . . . 6 ⊢ (𝑝 = ;;341 → (𝑝 ∈ ℙ ↔ ;;341 ∈ ℙ))
9	7, 8	imbi12d 344	. . . . 5 ⊢ (𝑝 = ;;341 → ((((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ) ↔ (((2↑;;341) mod ;;341) = (2 mod ;;341) → ;;341 ∈ ℙ)))
10	9	notbid 318	. . . 4 ⊢ (𝑝 = ;;341 → (¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ¬ (((2↑;;341) mod ;;341) = (2 mod ;;341) → ;;341 ∈ ℙ)))
11	2, 10	anbi12d 631	. . 3 ⊢ (𝑝 = ;;341 → ((𝑝 ∈ (ℤ_≥‘3) ∧ ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ)) ↔ (;;341 ∈ (ℤ_≥‘3) ∧ ¬ (((2↑;;341) mod ;;341) = (2 mod ;;341) → ;;341 ∈ ℙ))))
12		3z 12625	. . . . 5 ⊢ 3 ∈ ℤ
13		3nn0 12520	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
14		4nn0 12521	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
15	13, 14	deccl 12722	. . . . . . 7 ⊢ ;34 ∈ ℕ₀
16		1nn0 12518	. . . . . . 7 ⊢ 1 ∈ ℕ₀
17	15, 16	deccl 12722	. . . . . 6 ⊢ ;;341 ∈ ℕ₀
18	17	nn0zi 12617	. . . . 5 ⊢ ;;341 ∈ ℤ
19	13	dec0h 12729	. . . . . 6 ⊢ 3 = ;03
20		0nn0 12517	. . . . . . 7 ⊢ 0 ∈ ℕ₀
21		3re 12322	. . . . . . . 8 ⊢ 3 ∈ ℝ
22		9re 12341	. . . . . . . 8 ⊢ 9 ∈ ℝ
23		3lt9 12446	. . . . . . . 8 ⊢ 3 < 9
24	21, 22, 23	ltleii 11367	. . . . . . 7 ⊢ 3 ≤ 9
25		3nn 12321	. . . . . . . 8 ⊢ 3 ∈ ℕ
26		0re 11246	. . . . . . . . 9 ⊢ 0 ∈ ℝ
27		9pos 12355	. . . . . . . . 9 ⊢ 0 < 9
28	26, 22, 27	ltleii 11367	. . . . . . . 8 ⊢ 0 ≤ 9
29	25, 14, 20, 28	decltdi 12746	. . . . . . 7 ⊢ 0 < ;34
30	20, 15, 13, 16, 24, 29	decleh 12742	. . . . . 6 ⊢ ;03 ≤ ;;341
31	19, 30	eqbrtri 5169	. . . . 5 ⊢ 3 ≤ ;;341
32		eluz2 12858	. . . . 5 ⊢ (;;341 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ ;;341 ∈ ℤ ∧ 3 ≤ ;;341))
33	12, 18, 31, 32	mpbir3an 1339	. . . 4 ⊢ ;;341 ∈ (ℤ_≥‘3)
34		341fppr2 47074	. . . . . . 7 ⊢ ;;341 ∈ ( FPPr ‘2)
35		fpprwppr 47079	. . . . . . 7 ⊢ (;;341 ∈ ( FPPr ‘2) → ((2↑;;341) mod ;;341) = (2 mod ;;341))
36	34, 35	ax-mp 5	. . . . . 6 ⊢ ((2↑;;341) mod ;;341) = (2 mod ;;341)
37		11t31e341 47072	. . . . . . . 8 ⊢ (;11 · ;31) = ;;341
38	37	eqcomi 2737	. . . . . . 7 ⊢ ;;341 = (;11 · ;31)
39		2z 12624	. . . . . . . . 9 ⊢ 2 ∈ ℤ
40	16, 16	deccl 12722	. . . . . . . . . 10 ⊢ ;11 ∈ ℕ₀
41	40	nn0zi 12617	. . . . . . . . 9 ⊢ ;11 ∈ ℤ
42		2nn0 12519	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
43	42	dec0h 12729	. . . . . . . . . 10 ⊢ 2 = ;02
44		2re 12316	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
45		2lt9 12447	. . . . . . . . . . . 12 ⊢ 2 < 9
46	44, 22, 45	ltleii 11367	. . . . . . . . . . 11 ⊢ 2 ≤ 9
47		0lt1 11766	. . . . . . . . . . 11 ⊢ 0 < 1
48	20, 16, 42, 16, 46, 47	decleh 12742	. . . . . . . . . 10 ⊢ ;02 ≤ ;11
49	43, 48	eqbrtri 5169	. . . . . . . . 9 ⊢ 2 ≤ ;11
50		eluz2 12858	. . . . . . . . 9 ⊢ (;11 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ ;11 ∈ ℤ ∧ 2 ≤ ;11))
51	39, 41, 49, 50	mpbir3an 1339	. . . . . . . 8 ⊢ ;11 ∈ (ℤ_≥‘2)
52	13, 16	deccl 12722	. . . . . . . . . 10 ⊢ ;31 ∈ ℕ₀
53	52	nn0zi 12617	. . . . . . . . 9 ⊢ ;31 ∈ ℤ
54		3pos 12347	. . . . . . . . . . 11 ⊢ 0 < 3
55	20, 13, 42, 16, 46, 54	decleh 12742	. . . . . . . . . 10 ⊢ ;02 ≤ ;31
56	43, 55	eqbrtri 5169	. . . . . . . . 9 ⊢ 2 ≤ ;31
57		eluz2 12858	. . . . . . . . 9 ⊢ (;31 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
58	39, 53, 56, 57	mpbir3an 1339	. . . . . . . 8 ⊢ ;31 ∈ (ℤ_≥‘2)
59		nprm 16658	. . . . . . . 8 ⊢ ((;11 ∈ (ℤ_≥‘2) ∧ ;31 ∈ (ℤ_≥‘2)) → ¬ (;11 · ;31) ∈ ℙ)
60	51, 58, 59	mp2an 691	. . . . . . 7 ⊢ ¬ (;11 · ;31) ∈ ℙ
61	38, 60	eqneltri 2848	. . . . . 6 ⊢ ¬ ;;341 ∈ ℙ
62	36, 61	pm3.2i 470	. . . . 5 ⊢ (((2↑;;341) mod ;;341) = (2 mod ;;341) ∧ ¬ ;;341 ∈ ℙ)
63		annim 403	. . . . 5 ⊢ ((((2↑;;341) mod ;;341) = (2 mod ;;341) ∧ ¬ ;;341 ∈ ℙ) ↔ ¬ (((2↑;;341) mod ;;341) = (2 mod ;;341) → ;;341 ∈ ℙ))
64	62, 63	mpbi 229	. . . 4 ⊢ ¬ (((2↑;;341) mod ;;341) = (2 mod ;;341) → ;;341 ∈ ℙ)
65	33, 64	pm3.2i 470	. . 3 ⊢ (;;341 ∈ (ℤ_≥‘3) ∧ ¬ (((2↑;;341) mod ;;341) = (2 mod ;;341) → ;;341 ∈ ℙ))
66	1, 11, 65	ceqsexv2d 3526	. 2 ⊢ ∃𝑝(𝑝 ∈ (ℤ_≥‘3) ∧ ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ))
67		df-rex 3068	. 2 ⊢ (∃𝑝 ∈ (ℤ_≥‘3) ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ∃𝑝(𝑝 ∈ (ℤ_≥‘3) ∧ ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ)))
68	66, 67	mpbir 230	1 ⊢ ∃𝑝 ∈ (ℤ_≥‘3) ¬ (((2↑𝑝) mod 𝑝) = (2 mod 𝑝) → 𝑝 ∈ ℙ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∃wrex 3067 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 0cc0 11138 1c1 11139 · cmul 11143 ≤ cle 11279 2c2 12297 3c3 12298 4c4 12299 9c9 12304 ℤcz 12588 cdc 12707 ℤ_≥cuz 12852 mod cmo 13866 ↑cexp 14058 ℙcprime 16641 FPPr cfppr 47064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-prm 16642 df-fppr 47065

This theorem is referenced by: (None)

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