pcadd2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pcadd2		GIF version

Theorem pcadd2 13039

Description: The inequality of pcadd 13038 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

**Hypotheses**
Ref	Expression
pcadd2.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
pcadd2.2	⊢ (𝜑 → 𝐴 ∈ ℚ)
pcadd2.3	⊢ (𝜑 → 𝐵 ∈ ℚ)
pcadd2.4	⊢ (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵))

**Assertion**
Ref	Expression
pcadd2	⊢ (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)))

**Proof of Theorem pcadd2**
Step	Hyp	Ref	Expression
1		pcadd2.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		pcadd2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ)
3		pcxcl 13009	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
4	1, 2, 3	syl2anc 411	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℝ^*)
5		pcadd2.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℚ)
6		qaddcl 9967	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
7	2, 5, 6	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8		pcxcl 13009	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^*)
9	1, 7, 8	syl2anc 411	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^*)
10		pcxcl 13009	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈ ℝ^*)
11	1, 5, 10	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝐵) ∈ ℝ^*)
12		pcadd2.4	. . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵))
13	4, 11, 12	xrltled 10132	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
14	1, 2, 5, 13	pcadd 13038	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15		qnegcl 9968	. . . . 5 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
16	5, 15	syl 14	. . . 4 ⊢ (𝜑 → -𝐵 ∈ ℚ)
17		pcxqcl 13010	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞))
18		zq 9958	. . . . . . . . . . . . 13 ⊢ ((𝑃 pCnt 𝐴) ∈ ℤ → (𝑃 pCnt 𝐴) ∈ ℚ)
19	18	orim1i 768	. . . . . . . . . . . 12 ⊢ (((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞) → ((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞))
20	17, 19	syl 14	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞))
21	1, 2, 20	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞))
22		pcxqcl 13010	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞))
23		zq 9958	. . . . . . . . . . . . 13 ⊢ ((𝑃 pCnt 𝐵) ∈ ℤ → (𝑃 pCnt 𝐵) ∈ ℚ)
24	23	orim1i 768	. . . . . . . . . . . 12 ⊢ (((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞) → ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞))
25	22, 24	syl 14	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞))
26	1, 5, 25	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞))
27		xqltnle 10627	. . . . . . . . . 10 ⊢ ((((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞) ∧ ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞)) → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
28	21, 26, 27	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
29	12, 28	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
30	1	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ)
31	16	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ)
32	7	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ)
33		pcneg 13023	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
34	1, 5, 33	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
35	34	breq1d 4119	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
36	35	biimpar 297	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
37	30, 31, 32, 36	pcadd 13038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))
38	37	ex 115	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))))
39		qcn 9966	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
40	5, 39	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41	40	negcld 8571	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝐵 ∈ ℂ)
42		qcn 9966	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
43	2, 42	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	41, 43, 40	add12d 8440	. . . . . . . . . . . 12 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
45	41, 40	addcomd 8424	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
46	40	negidd 8574	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + -𝐵) = 0)
47	45, 46	eqtrd 2265	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝐵 + 𝐵) = 0)
48	47	oveq2d 6066	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
49	43	addridd 8422	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
50	44, 48, 49	3eqtrd 2269	. . . . . . . . . . 11 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
51	50	oveq2d 6066	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
52	34, 51	breq12d 4122	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
53	38, 52	sylibd 149	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
54	29, 53	mtod 669	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
55		pcxqcl 13010	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
56		zq 9958	. . . . . . . . . . 11 ⊢ ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ)
57	56	orim1i 768	. . . . . . . . . 10 ⊢ (((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
58	55, 57	syl 14	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
59	1, 7, 58	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
60		xqltnle 10627	. . . . . . . 8 ⊢ ((((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞) ∧ ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞)) → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
61	59, 26, 60	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
62	54, 61	mpbird 167	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵))
63	9, 11, 62	xrltled 10132	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
64	63, 34	breqtrrd 4137	. . . 4 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
65	1, 7, 16, 64	pcadd 13038	. . 3 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
66	43, 40, 41	addassd 8296	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
67	46	oveq2d 6066	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
68	66, 67, 49	3eqtrd 2269	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
69	68	oveq2d 6066	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
70	65, 69	breqtrd 4135	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
71	4, 9, 14, 70	xrletrid 10138	1 ⊢ (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 = wceq 1398 ∈ wcel 2203 class class class wbr 4109 (class class class)co 6050 ℂcc 8125 0cc0 8127 + caddc 8130 +∞cpnf 8305 ℝ^*cxr 8307 < clt 8308 ≤ cle 8309 -cneg 8445 ℤcz 9577 ℚcq 9951 ℙcprime 12804 pCnt cpc 12982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-1o 6647 df-2o 6648 df-er 6767 df-en 6976 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-fz 10343 df-fzo 10477 df-fl 10630 df-mod 10685 df-seqfrec 10810 df-exp 10901 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-dvds 12474 df-gcd 12650 df-prm 12805 df-pc 12983

This theorem is referenced by: (None)

W3C validator