pcadd2 - Intuitionistic Logic Explorer

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Theorem pcadd2 12606

Description: The inequality of pcadd 12605 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 12576	. . 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 9755	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
7	2, 5, 6	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8		pcxcl 12576	. . 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 12576	. . . . 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 9920	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
14	1, 2, 5, 13	pcadd 12605	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15		qnegcl 9756	. . . . 5 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
16	5, 15	syl 14	. . . 4 ⊢ (𝜑 → -𝐵 ∈ ℚ)
17		pcxqcl 12577	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞))
18		zq 9746	. . . . . . . . . . . . 13 ⊢ ((𝑃 pCnt 𝐴) ∈ ℤ → (𝑃 pCnt 𝐴) ∈ ℚ)
19	18	orim1i 761	. . . . . . . . . . . 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 12577	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞))
23		zq 9746	. . . . . . . . . . . . 13 ⊢ ((𝑃 pCnt 𝐵) ∈ ℤ → (𝑃 pCnt 𝐵) ∈ ℚ)
24	23	orim1i 761	. . . . . . . . . . . 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 10408	. . . . . . . . . 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 12590	. . . . . . . . . . . . . 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 4053	. . . . . . . . . . . 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 12605	. . . . . . . . . 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 9754	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
40	5, 39	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41	40	negcld 8369	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝐵 ∈ ℂ)
42		qcn 9754	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
43	2, 42	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	41, 43, 40	add12d 8238	. . . . . . . . . . . 12 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
45	41, 40	addcomd 8222	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
46	40	negidd 8372	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + -𝐵) = 0)
47	45, 46	eqtrd 2237	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝐵 + 𝐵) = 0)
48	47	oveq2d 5959	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
49	43	addridd 8220	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
50	44, 48, 49	3eqtrd 2241	. . . . . . . . . . 11 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
51	50	oveq2d 5959	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
52	34, 51	breq12d 4056	. . . . . . . . 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 664	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
55		pcxqcl 12577	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
56		zq 9746	. . . . . . . . . . 11 ⊢ ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ)
57	56	orim1i 761	. . . . . . . . . 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 10408	. . . . . . . 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 9920	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
64	63, 34	breqtrrd 4071	. . . 4 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
65	1, 7, 16, 64	pcadd 12605	. . 3 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
66	43, 40, 41	addassd 8094	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
67	46	oveq2d 5959	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
68	66, 67, 49	3eqtrd 2241	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
69	68	oveq2d 5959	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
70	65, 69	breqtrd 4069	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
71	4, 9, 14, 70	xrletrid 9926	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 709 = wceq 1372 ∈ wcel 2175 class class class wbr 4043 (class class class)co 5943 ℂcc 7922 0cc0 7924 + caddc 7927 +∞cpnf 8103 ℝ^*cxr 8105 < clt 8106 ≤ cle 8107 -cneg 8243 ℤcz 9371 ℚcq 9739 ℙcprime 12371 pCnt cpc 12549

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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-isom 5279 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-frec 6476 df-1o 6501 df-2o 6502 df-er 6619 df-en 6827 df-sup 7085 df-inf 7086 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-fz 10130 df-fzo 10264 df-fl 10411 df-mod 10466 df-seqfrec 10591 df-exp 10682 df-cj 11095 df-re 11096 df-im 11097 df-rsqrt 11251 df-abs 11252 df-dvds 12041 df-gcd 12217 df-prm 12372 df-pc 12550

This theorem is referenced by: (None)

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