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Theorem pcadd2 16226

Description: The inequality of pcadd 16225 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 16197	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
4	1, 2, 3	syl2anc 586	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℝ^*)
5		pcadd2.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℚ)
6		qaddcl 12365	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
7	2, 5, 6	syl2anc 586	. . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8		pcxcl 16197	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^*)
9	1, 7, 8	syl2anc 586	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^*)
10		pcxcl 16197	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈ ℝ^*)
11	1, 5, 10	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝐵) ∈ ℝ^*)
12		pcadd2.4	. . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵))
13	4, 11, 12	xrltled 12544	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
14	1, 2, 5, 13	pcadd 16225	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15		qnegcl 12366	. . . . 5 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → -𝐵 ∈ ℚ)
17		xrltnle 10708	. . . . . . . . . 10 ⊢ (((𝑃 pCnt 𝐴) ∈ ℝ^* ∧ (𝑃 pCnt 𝐵) ∈ ℝ^*) → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
18	4, 11, 17	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
19	12, 18	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
20	1	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ)
21	16	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ)
22	7	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ)
23		pcneg 16210	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
24	1, 5, 23	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
25	24	breq1d 5076	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
26	25	biimpar 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
27	20, 21, 22, 26	pcadd 16225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))
28	27	ex 415	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))))
29		qcn 12363	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
30	5, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
31	30	negcld 10984	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝐵 ∈ ℂ)
32		qcn 12363	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
33	2, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
34	31, 33, 30	add12d 10866	. . . . . . . . . . . 12 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
35	31, 30	addcomd 10842	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
36	30	negidd 10987	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + -𝐵) = 0)
37	35, 36	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝐵 + 𝐵) = 0)
38	37	oveq2d 7172	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
39	33	addid1d 10840	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
40	34, 38, 39	3eqtrd 2860	. . . . . . . . . . 11 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
41	40	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
42	24, 41	breq12d 5079	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
43	28, 42	sylibd 241	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
44	19, 43	mtod 200	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
45		xrltnle 10708	. . . . . . . 8 ⊢ (((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^* ∧ (𝑃 pCnt 𝐵) ∈ ℝ^*) → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
46	9, 11, 45	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
47	44, 46	mpbird 259	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵))
48	9, 11, 47	xrltled 12544	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
49	48, 24	breqtrrd 5094	. . . 4 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
50	1, 7, 16, 49	pcadd 16225	. . 3 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
51	33, 30, 31	addassd 10663	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
52	36	oveq2d 7172	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
53	51, 52, 39	3eqtrd 2860	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
54	53	oveq2d 7172	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
55	50, 54	breqtrd 5092	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
56	4, 9, 14, 55	xrletrid 12549	1 ⊢ (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 0cc0 10537 + caddc 10540 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 -cneg 10871 ℚcq 12349 ℙcprime 16015 pCnt cpc 16173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 df-pc 16174

This theorem is referenced by: sylow1lem1 18723

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