pcadd2 - Intuitionistic Logic Explorer

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Theorem pcadd2 13064

Description: The inequality of pcadd 13063 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 13034	. . 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 9985	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
7	2, 5, 6	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8		pcxcl 13034	. . 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 13034	. . . . 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 10151	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
14	1, 2, 5, 13	pcadd 13063	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15		qnegcl 9986	. . . . 5 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
16	5, 15	syl 14	. . . 4 ⊢ (𝜑 → -𝐵 ∈ ℚ)
17		pcxqcl 13035	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞))
18		zq 9976	. . . . . . . . . . . . 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 13035	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞))
23		zq 9976	. . . . . . . . . . . . 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 10651	. . . . . . . . . 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 13048	. . . . . . . . . . . . . 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 4124	. . . . . . . . . . . 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 13063	. . . . . . . . . 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 9984	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
40	5, 39	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41	40	negcld 8587	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝐵 ∈ ℂ)
42		qcn 9984	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
43	2, 42	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	41, 43, 40	add12d 8456	. . . . . . . . . . . 12 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
45	41, 40	addcomd 8440	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
46	40	negidd 8590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + -𝐵) = 0)
47	45, 46	eqtrd 2267	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝐵 + 𝐵) = 0)
48	47	oveq2d 6074	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
49	43	addridd 8438	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
50	44, 48, 49	3eqtrd 2271	. . . . . . . . . . 11 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
51	50	oveq2d 6074	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
52	34, 51	breq12d 4127	. . . . . . . . 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 13035	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
56		zq 9976	. . . . . . . . . . 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 10651	. . . . . . . 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 10151	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
64	63, 34	breqtrrd 4142	. . . 4 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
65	1, 7, 16, 64	pcadd 13063	. . 3 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
66	43, 40, 41	addassd 8312	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
67	46	oveq2d 6074	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
68	66, 67, 49	3eqtrd 2271	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
69	68	oveq2d 6074	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
70	65, 69	breqtrd 4140	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
71	4, 9, 14, 70	xrletrid 10157	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 2205 class class class wbr 4114 (class class class)co 6058 ℂcc 8141 0cc0 8143 + caddc 8146 +∞cpnf 8321 ℝ^*cxr 8323 < clt 8324 ≤ cle 8325 -cneg 8461 ℤcz 9594 ℚcq 9969 ℙcprime 12829 pCnt cpc 13007

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-2o 6661 df-er 6780 df-en 6989 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-dvds 12499 df-gcd 12675 df-prm 12830 df-pc 13008

This theorem is referenced by: (None)

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