prmlem1a - Intuitionistic Logic Explorer

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Theorem prmlem1a 13250

Description: Lemma for prmlem1 13251 and prmlem2 13263. (Contributed by Mario Carneiro, 18-Feb-2014.)

**Hypotheses**
Ref	Expression
prmlem1.n	⊢ 𝑁 ∈ ℕ
prmlem1.gt	⊢ 1 < 𝑁
prmlem1.2	⊢ ¬ 2 ∥ 𝑁
prmlem1.3	⊢ ¬ 3 ∥ 𝑁
prmlem1a.x	⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ_≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))

**Assertion**
Ref	Expression
prmlem1a	⊢ 𝑁 ∈ ℙ

Distinct variable group: 𝑥,𝑁

**Proof of Theorem prmlem1a**
Step	Hyp	Ref	Expression
1		prmlem1.n	. . 3 ⊢ 𝑁 ∈ ℕ
2		prmlem1.gt	. . 3 ⊢ 1 < 𝑁
3		eluz2b2 10019	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
4	1, 2, 3	mpbir2an 955	. 2 ⊢ 𝑁 ∈ (ℤ_≥‘2)
5		prmlem1.2	. . . . . 6 ⊢ ¬ 2 ∥ 𝑁
6	5	2a1i 27	. . . . 5 ⊢ (𝑥 ∈ ℙ → ((𝑥↑2) ≤ 𝑁 → ¬ 2 ∥ 𝑁))
7		breq1 4134	. . . . . . 7 ⊢ (𝑥 = 2 → (𝑥 ∥ 𝑁 ↔ 2 ∥ 𝑁))
8	7	notbid 677	. . . . . 6 ⊢ (𝑥 = 2 → (¬ 𝑥 ∥ 𝑁 ↔ ¬ 2 ∥ 𝑁))
9	8	imbi2d 230	. . . . 5 ⊢ (𝑥 = 2 → (((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁) ↔ ((𝑥↑2) ≤ 𝑁 → ¬ 2 ∥ 𝑁)))
10	6, 9	syl5ibrcom 157	. . . 4 ⊢ (𝑥 ∈ ℙ → (𝑥 = 2 → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)))
11		prmnn 12913	. . . . . . 7 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ)
12	11	adantr 276	. . . . . 6 ⊢ ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → 𝑥 ∈ ℕ)
13		eldifsn 3841	. . . . . . 7 ⊢ (𝑥 ∈ (ℙ ∖ {2}) ↔ (𝑥 ∈ ℙ ∧ 𝑥 ≠ 2))
14		n2dvds1 12704	. . . . . . . . . 10 ⊢ ¬ 2 ∥ 1
15		prmlem1a.x	. . . . . . . . . . . 12 ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ_≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
16		prmlem1.3	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 𝑁
17	16	a1i 9	. . . . . . . . . . . 12 ⊢ (3 ∈ ℙ → ¬ 3 ∥ 𝑁)
18		3p2e5 9455	. . . . . . . . . . . 12 ⊢ (3 + 2) = 5
19	15, 17, 18	prmlem0 13249	. . . . . . . . . . 11 ⊢ ((¬ 2 ∥ 3 ∧ 𝑥 ∈ (ℤ_≥‘3)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
20		1nprm 12917	. . . . . . . . . . . 12 ⊢ ¬ 1 ∈ ℙ
21	20	pm2.21i 655	. . . . . . . . . . 11 ⊢ (1 ∈ ℙ → ¬ 1 ∥ 𝑁)
22		1p2e3 9448	. . . . . . . . . . 11 ⊢ (1 + 2) = 3
23	19, 21, 22	prmlem0 13249	. . . . . . . . . 10 ⊢ ((¬ 2 ∥ 1 ∧ 𝑥 ∈ (ℤ_≥‘1)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
24	14, 23	mpan 428	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℤ_≥‘1) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
25		nnuz 9974	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
26	24, 25	eleq2s 2333	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
27	26	expd 258	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑥 ∈ (ℙ ∖ {2}) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)))
28	13, 27	biimtrrid 153	. . . . . 6 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)))
29	12, 28	mpcom 36	. . . . 5 ⊢ ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))
30	29	ex 115	. . . 4 ⊢ (𝑥 ∈ ℙ → (𝑥 ≠ 2 → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)))
31		prmz 12914	. . . . . 6 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℤ)
32		2z 9683	. . . . . 6 ⊢ 2 ∈ ℤ
33		zdceq 9731	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑥 = 2)
34	31, 32, 33	sylancl 417	. . . . 5 ⊢ (𝑥 ∈ ℙ → DECID 𝑥 = 2)
35		dcne 2431	. . . . 5 ⊢ (DECID 𝑥 = 2 ↔ (𝑥 = 2 ∨ 𝑥 ≠ 2))
36	34, 35	sylib 122	. . . 4 ⊢ (𝑥 ∈ ℙ → (𝑥 = 2 ∨ 𝑥 ≠ 2))
37	10, 30, 36	mpjaod 730	. . 3 ⊢ (𝑥 ∈ ℙ → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))
38	37	rgen 2603	. 2 ⊢ ∀𝑥 ∈ ℙ ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)
39		isprm5 12946	. 2 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ_≥‘2) ∧ ∀𝑥 ∈ ℙ ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)))
40	4, 38, 39	mpbir2an 955	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 720 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 ∖ cdif 3217 {csn 3709 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 1c1 8186 < clt 8366 ≤ cle 8367 ℕcn 9313 2c2 9364 3c3 9365 5c5 9367 ℤcz 9655 ℤ_≥cuz 9937 ↑cexp 10996 ∥ cdvds 12579 ℙcprime 12910

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-1o 6691 df-2o 6692 df-er 6811 df-en 7027 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-dvds 12580 df-prm 12911

This theorem is used by: prmlem1 13251 prmlem2 13263

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