pcadd2 - Intuitionistic Logic Explorer

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

Theorem pcadd2 12904

Description: The inequality of pcadd 12903 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 12874	. . 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 9859	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
7	2, 5, 6	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8		pcxcl 12874	. . 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 12874	. . . . 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 10024	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
14	1, 2, 5, 13	pcadd 12903	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15		qnegcl 9860	. . . . 5 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
16	5, 15	syl 14	. . . 4 ⊢ (𝜑 → -𝐵 ∈ ℚ)
17		pcxqcl 12875	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞))
18		zq 9850	. . . . . . . . . . . . 13 ⊢ ((𝑃 pCnt 𝐴) ∈ ℤ → (𝑃 pCnt 𝐴) ∈ ℚ)
19	18	orim1i 765	. . . . . . . . . . . 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 12875	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞))
23		zq 9850	. . . . . . . . . . . . 13 ⊢ ((𝑃 pCnt 𝐵) ∈ ℤ → (𝑃 pCnt 𝐵) ∈ ℚ)
24	23	orim1i 765	. . . . . . . . . . . 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 10517	. . . . . . . . . 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 12888	. . . . . . . . . . . . . 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 4096	. . . . . . . . . . . 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 12903	. . . . . . . . . 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 9858	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
40	5, 39	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41	40	negcld 8467	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝐵 ∈ ℂ)
42		qcn 9858	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
43	2, 42	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	41, 43, 40	add12d 8336	. . . . . . . . . . . 12 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
45	41, 40	addcomd 8320	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
46	40	negidd 8470	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + -𝐵) = 0)
47	45, 46	eqtrd 2262	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝐵 + 𝐵) = 0)
48	47	oveq2d 6029	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
49	43	addridd 8318	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
50	44, 48, 49	3eqtrd 2266	. . . . . . . . . . 11 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
51	50	oveq2d 6029	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
52	34, 51	breq12d 4099	. . . . . . . . 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 667	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
55		pcxqcl 12875	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
56		zq 9850	. . . . . . . . . . 11 ⊢ ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ)
57	56	orim1i 765	. . . . . . . . . 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 10517	. . . . . . . 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 10024	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
64	63, 34	breqtrrd 4114	. . . 4 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
65	1, 7, 16, 64	pcadd 12903	. . 3 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
66	43, 40, 41	addassd 8192	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
67	46	oveq2d 6029	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
68	66, 67, 49	3eqtrd 2266	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
69	68	oveq2d 6029	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
70	65, 69	breqtrd 4112	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
71	4, 9, 14, 70	xrletrid 10030	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 713 = wceq 1395 ∈ wcel 2200 class class class wbr 4086 (class class class)co 6013 ℂcc 8020 0cc0 8022 + caddc 8025 +∞cpnf 8201 ℝ^*cxr 8203 < clt 8204 ≤ cle 8205 -cneg 8341 ℤcz 9469 ℚcq 9843 ℙcprime 12669 pCnt cpc 12847

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-1o 6577 df-2o 6578 df-er 6697 df-en 6905 df-sup 7174 df-inf 7175 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-n0 9393 df-z 9470 df-uz 9746 df-q 9844 df-rp 9879 df-fz 10234 df-fzo 10368 df-fl 10520 df-mod 10575 df-seqfrec 10700 df-exp 10791 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-dvds 12339 df-gcd 12515 df-prm 12670 df-pc 12848

This theorem is referenced by: (None)

W3C validator