prmlem1 - Intuitionistic Logic Explorer

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Theorem prmlem1 13251

Description: A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (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
prmlem1.lt	⊢ 𝑁 < ;25

**Assertion**
Ref	Expression
prmlem1	⊢ 𝑁 ∈ ℙ

Proof of Theorem prmlem1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmlem1.n	. 2 ⊢ 𝑁 ∈ ℕ
2		prmlem1.gt	. 2 ⊢ 1 < 𝑁
3		prmlem1.2	. 2 ⊢ ¬ 2 ∥ 𝑁
4		prmlem1.3	. 2 ⊢ ¬ 3 ∥ 𝑁
5		eluzelre 9948	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ_≥‘5) → 𝑥 ∈ ℝ)
6	5	resqcld 11158	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ_≥‘5) → (𝑥↑2) ∈ ℝ)
7		eluzle 9950	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ_≥‘5) → 5 ≤ 𝑥)
8		5re 9392	. . . . . . . . 9 ⊢ 5 ∈ ℝ
9		5nn0 9594	. . . . . . . . . 10 ⊢ 5 ∈ ℕ₀
10	9	nn0ge0i 9601	. . . . . . . . 9 ⊢ 0 ≤ 5
11		le2sq2 11073	. . . . . . . . 9 ⊢ (((5 ∈ ℝ ∧ 0 ≤ 5) ∧ (𝑥 ∈ ℝ ∧ 5 ≤ 𝑥)) → (5↑2) ≤ (𝑥↑2))
12	8, 10, 11	mpanl12 440	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 5 ≤ 𝑥) → (5↑2) ≤ (𝑥↑2))
13	5, 7, 12	syl2anc 415	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ_≥‘5) → (5↑2) ≤ (𝑥↑2))
14	1	nnrei 9322	. . . . . . . 8 ⊢ 𝑁 ∈ ℝ
15	8	resqcli 11082	. . . . . . . 8 ⊢ (5↑2) ∈ ℝ
16		prmlem1.lt	. . . . . . . . . 10 ⊢ 𝑁 < ;25
17		5cn 9393	. . . . . . . . . . . 12 ⊢ 5 ∈ ℂ
18	17	sqvali 11077	. . . . . . . . . . 11 ⊢ (5↑2) = (5 · 5)
19		5t5e25 9895	. . . . . . . . . . 11 ⊢ (5 · 5) = ;25
20	18, 19	eqtri 2259	. . . . . . . . . 10 ⊢ (5↑2) = ;25
21	16, 20	breqtrri 4158	. . . . . . . . 9 ⊢ 𝑁 < (5↑2)
22		ltletr 8421	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑁 < (5↑2) ∧ (5↑2) ≤ (𝑥↑2)) → 𝑁 < (𝑥↑2)))
23	21, 22	mpani 434	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2)))
24	14, 15, 23	mp3an12 1368	. . . . . . 7 ⊢ ((𝑥↑2) ∈ ℝ → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2)))
25	6, 13, 24	sylc 62	. . . . . 6 ⊢ (𝑥 ∈ (ℤ_≥‘5) → 𝑁 < (𝑥↑2))
26	1	nnzi 9676	. . . . . . 7 ⊢ 𝑁 ∈ ℤ
27		eluzelz 9947	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ_≥‘5) → 𝑥 ∈ ℤ)
28		zsqcl 11068	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ)
29	27, 28	syl 14	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ_≥‘5) → (𝑥↑2) ∈ ℤ)
30		zltnle 9701	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ (𝑥↑2) ∈ ℤ) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁))
31	26, 29, 30	sylancr 418	. . . . . 6 ⊢ (𝑥 ∈ (ℤ_≥‘5) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁))
32	25, 31	mpbid 147	. . . . 5 ⊢ (𝑥 ∈ (ℤ_≥‘5) → ¬ (𝑥↑2) ≤ 𝑁)
33	32	pm2.21d 628	. . . 4 ⊢ (𝑥 ∈ (ℤ_≥‘5) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))
34	33	adantld 278	. . 3 ⊢ (𝑥 ∈ (ℤ_≥‘5) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
35	34	adantl 277	. 2 ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ_≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁))
36	1, 2, 3, 4, 35	prmlem1a 13250	1 ⊢ 𝑁 ∈ ℙ

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 ∈ wcel 2209 ∖ cdif 3217 {csn 3709 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 ℝcr 8184 0cc0 8185 1c1 8186 · cmul 8190 < clt 8366 ≤ cle 8367 ℕcn 9313 2c2 9364 3c3 9365 5c5 9367 ℤcz 9655 cdc 9788 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-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-z 9656 df-dec 9789 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: 5prm 13252 7prm 13254 11prm 13258 13prm 13259 17prm 13260 19prm 13261 23prm 13262

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