ppiqeq0 - Intuitionistic Logic Explorer

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Theorem ppiqeq0 16280

Description: The prime-counting function π is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Assertion**
Ref	Expression
ppiqeq0	⊢ (𝐴 ∈ ℚ → ((π‘𝐴) = 0 ↔ 𝐴 < 2))

**Proof of Theorem ppiqeq0**
Step	Hyp	Ref	Expression
1		2z 9683	. . . . 5 ⊢ 2 ∈ ℤ
2		zq 10042	. . . . 5 ⊢ (2 ∈ ℤ → 2 ∈ ℚ)
3	1, 2	ax-mp 5	. . . 4 ⊢ 2 ∈ ℚ
4		qdclt 10697	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 2 ∈ ℚ) → DECID 𝐴 < 2)
5	3, 4	mpan2 429	. . 3 ⊢ (𝐴 ∈ ℚ → DECID 𝐴 < 2)
6		2re 9383	. . . . . 6 ⊢ 2 ∈ ℝ
7		qre 10041	. . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
8		lenlt 8407	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
9	6, 7, 8	sylancr 418	. . . . 5 ⊢ (𝐴 ∈ ℚ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
10		ppiqnncl 16278	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ)
11	10	nnne0d 9358	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 2 ≤ 𝐴) → (π‘𝐴) ≠ 0)
12	11	ex 115	. . . . 5 ⊢ (𝐴 ∈ ℚ → (2 ≤ 𝐴 → (π‘𝐴) ≠ 0))
13	9, 12	sylbird 170	. . . 4 ⊢ (𝐴 ∈ ℚ → (¬ 𝐴 < 2 → (π‘𝐴) ≠ 0))
14		df-ne 2421	. . . 4 ⊢ ((π‘𝐴) ≠ 0 ↔ ¬ (π‘𝐴) = 0)
15	13, 14	imbitrdi 161	. . 3 ⊢ (𝐴 ∈ ℚ → (¬ 𝐴 < 2 → ¬ (π‘𝐴) = 0))
16		condc 865	. . 3 ⊢ (DECID 𝐴 < 2 → ((¬ 𝐴 < 2 → ¬ (π‘𝐴) = 0) → ((π‘𝐴) = 0 → 𝐴 < 2)))
17	5, 15, 16	sylc 62	. 2 ⊢ (𝐴 ∈ ℚ → ((π‘𝐴) = 0 → 𝐴 < 2))
18		flqcl 10725	. . . . . . . 8 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
19		zq 10042	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ)
20	18, 19	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℚ)
21	20	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℚ)
22		1z 9681	. . . . . . . 8 ⊢ 1 ∈ ℤ
23		zq 10042	. . . . . . . 8 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ ℚ
25	24	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → 1 ∈ ℚ)
26		flqlt 10738	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℚ ∧ 2 ∈ ℤ) → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
27	1, 26	mpan2 429	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
28	27	biimpa 296	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (⌊‘𝐴) < 2)
29		df-2 9372	. . . . . . . 8 ⊢ 2 = (1 + 1)
30	28, 29	breqtrdi 4172	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1))
31	18	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ)
32		zleltp1 9711	. . . . . . . 8 ⊢ (((⌊‘𝐴) ∈ ℤ ∧ 1 ∈ ℤ) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1)))
33	31, 22, 32	sylancl 417	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1)))
34	30, 33	mpbird 167	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (⌊‘𝐴) ≤ 1)
35		ppiqwordi 16268	. . . . . 6 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 1 ∈ ℚ ∧ (⌊‘𝐴) ≤ 1) → (π‘(⌊‘𝐴)) ≤ (π‘1))
36	21, 25, 34, 35	syl3anc 1278	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (π‘(⌊‘𝐴)) ≤ (π‘1))
37		ppiqfl 16266	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (π‘(⌊‘𝐴)) = (π‘𝐴))
38	37	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (π‘(⌊‘𝐴)) = (π‘𝐴))
39		ppi1 16270	. . . . . 6 ⊢ (π‘1) = 0
40	39	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (π‘1) = 0)
41	36, 38, 40	3brtr3d 4162	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (π‘𝐴) ≤ 0)
42		ppiqcl 16251	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (π‘𝐴) ∈ ℕ₀)
43	42	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (π‘𝐴) ∈ ℕ₀)
44		nn0le0eq0 9602	. . . . 5 ⊢ ((π‘𝐴) ∈ ℕ₀ → ((π‘𝐴) ≤ 0 ↔ (π‘𝐴) = 0))
45	43, 44	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → ((π‘𝐴) ≤ 0 ↔ (π‘𝐴) = 0))
46	41, 45	mpbid 147	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 < 2) → (π‘𝐴) = 0)
47	46	ex 115	. 2 ⊢ (𝐴 ∈ ℚ → (𝐴 < 2 → (π‘𝐴) = 0))
48	17, 47	impbid 129	1 ⊢ (𝐴 ∈ ℚ → ((π‘𝐴) = 0 ↔ 𝐴 < 2))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 ℝcr 8184 0cc0 8185 1c1 8186 + caddc 8188 < clt 8366 ≤ cle 8367 2c2 9364 ℕ₀cn0 9574 ℤcz 9655 ℚcq 10035 ⌊cfl 10720 πcppi 16234

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-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-irdg 6645 df-frec 6666 df-1o 6691 df-2o 6692 df-oadd 6695 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 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-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-icc 10314 df-fz 10429 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-ihash 11237 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-dvds 12580 df-prm 12911 df-ppi 16237

This theorem is used by: ppiqltx 16281

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