pcadd2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pcadd2		Structured version Visualization version GIF version

Theorem pcadd2 16924

Description: The inequality of pcadd 16923 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 16895	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℝ^*)
5		pcadd2.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℚ)
6		qaddcl 13005	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
7	2, 5, 6	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8		pcxcl 16895	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^*)
9	1, 7, 8	syl2anc 584	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ^*)
10		pcxcl 16895	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈ ℝ^*)
11	1, 5, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝐵) ∈ ℝ^*)
12		pcadd2.4	. . . 4 ⊢ (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵))
13	4, 11, 12	xrltled 13189	. . 3 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
14	1, 2, 5, 13	pcadd 16923	. 2 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15		qnegcl 13006	. . . . 5 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
16	5, 15	syl 17	. . . 4 ⊢ (𝜑 → -𝐵 ∈ ℚ)
17		xrltnle 11326	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
19	12, 18	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
20	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ)
21	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ)
22	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ)
23		pcneg 16908	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
24	1, 5, 23	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
25	24	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
26	25	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
27	20, 21, 22, 26	pcadd 16923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))
28	27	ex 412	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))))
29		qcn 13003	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
30	5, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
31	30	negcld 11605	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝐵 ∈ ℂ)
32		qcn 13003	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
33	2, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
34	31, 33, 30	add12d 11486	. . . . . . . . . . . 12 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
35	31, 30	addcomd 11461	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
36	30	negidd 11608	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + -𝐵) = 0)
37	35, 36	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝐵 + 𝐵) = 0)
38	37	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
39	33	addridd 11459	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
40	34, 38, 39	3eqtrd 2779	. . . . . . . . . . 11 ⊢ (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
41	40	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
42	24, 41	breq12d 5161	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
43	28, 42	sylibd 239	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
44	19, 43	mtod 198	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
45		xrltnle 11326	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
47	44, 46	mpbird 257	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵))
48	9, 11, 47	xrltled 13189	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
49	48, 24	breqtrrd 5176	. . . 4 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
50	1, 7, 16, 49	pcadd 16923	. . 3 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
51	33, 30, 31	addassd 11281	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
52	36	oveq2d 7447	. . . . 5 ⊢ (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
53	51, 52, 39	3eqtrd 2779	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
54	53	oveq2d 7447	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
55	50, 54	breqtrd 5174	. 2 ⊢ (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
56	4, 9, 14, 55	xrletrid 13194	1 ⊢ (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 0cc0 11153 + caddc 11156 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 -cneg 11491 ℚcq 12988 ℙcprime 16705 pCnt cpc 16870

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871

This theorem is referenced by: sylow1lem1 19631

W3C validator