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Theorem pcadd2 12479
Description: The inequality of pcadd 12478 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 (𝜑𝑃 ∈ ℙ)
pcadd2.2 (𝜑𝐴 ∈ ℚ)
pcadd2.3 (𝜑𝐵 ∈ 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
StepHypRef Expression
1 pcadd2.1 . . 3 (𝜑𝑃 ∈ ℙ)
2 pcadd2.2 . . 3 (𝜑𝐴 ∈ ℚ)
3 pcxcl 12449 . . 3 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ*)
41, 2, 3syl2anc 411 . 2 (𝜑 → (𝑃 pCnt 𝐴) ∈ ℝ*)
5 pcadd2.3 . . . 4 (𝜑𝐵 ∈ ℚ)
6 qaddcl 9700 . . . 4 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
72, 5, 6syl2anc 411 . . 3 (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
8 pcxcl 12449 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ*)
91, 7, 8syl2anc 411 . 2 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ*)
10 pcxcl 12449 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈ ℝ*)
111, 5, 10syl2anc 411 . . . 4 (𝜑 → (𝑃 pCnt 𝐵) ∈ ℝ*)
12 pcadd2.4 . . . 4 (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵))
134, 11, 12xrltled 9865 . . 3 (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
141, 2, 5, 13pcadd 12478 . 2 (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
15 qnegcl 9701 . . . . 5 (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
165, 15syl 14 . . . 4 (𝜑 → -𝐵 ∈ ℚ)
17 pcxqcl 12450 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞))
18 zq 9691 . . . . . . . . . . . . 13 ((𝑃 pCnt 𝐴) ∈ ℤ → (𝑃 pCnt 𝐴) ∈ ℚ)
1918orim1i 761 . . . . . . . . . . . 12 (((𝑃 pCnt 𝐴) ∈ ℤ ∨ (𝑃 pCnt 𝐴) = +∞) → ((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞))
2017, 19syl 14 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → ((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞))
211, 2, 20syl2anc 411 . . . . . . . . . 10 (𝜑 → ((𝑃 pCnt 𝐴) ∈ ℚ ∨ (𝑃 pCnt 𝐴) = +∞))
22 pcxqcl 12450 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞))
23 zq 9691 . . . . . . . . . . . . 13 ((𝑃 pCnt 𝐵) ∈ ℤ → (𝑃 pCnt 𝐵) ∈ ℚ)
2423orim1i 761 . . . . . . . . . . . 12 (((𝑃 pCnt 𝐵) ∈ ℤ ∨ (𝑃 pCnt 𝐵) = +∞) → ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞))
2522, 24syl 14 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞))
261, 5, 25syl2anc 411 . . . . . . . . . 10 (𝜑 → ((𝑃 pCnt 𝐵) ∈ ℚ ∨ (𝑃 pCnt 𝐵) = +∞))
27 xqltnle 10336 . . . . . . . . . 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)))
2821, 26, 27syl2anc 411 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
2912, 28mpbid 147 . . . . . . . 8 (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
301adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ)
3116adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ)
327adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ)
33 pcneg 12463 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
341, 5, 33syl2anc 411 . . . . . . . . . . . . 13 (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
3534breq1d 4039 . . . . . . . . . . . 12 (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
3635biimpar 297 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
3730, 31, 32, 36pcadd 12478 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))
3837ex 115 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))))
39 qcn 9699 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
405, 39syl 14 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ ℂ)
4140negcld 8317 . . . . . . . . . . . . 13 (𝜑 → -𝐵 ∈ ℂ)
42 qcn 9699 . . . . . . . . . . . . . 14 (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
432, 42syl 14 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
4441, 43, 40add12d 8186 . . . . . . . . . . . 12 (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
4541, 40addcomd 8170 . . . . . . . . . . . . . 14 (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
4640negidd 8320 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 + -𝐵) = 0)
4745, 46eqtrd 2226 . . . . . . . . . . . . 13 (𝜑 → (-𝐵 + 𝐵) = 0)
4847oveq2d 5934 . . . . . . . . . . . 12 (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
4943addridd 8168 . . . . . . . . . . . 12 (𝜑 → (𝐴 + 0) = 𝐴)
5044, 48, 493eqtrd 2230 . . . . . . . . . . 11 (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
5150oveq2d 5934 . . . . . . . . . 10 (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
5234, 51breq12d 4042 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
5338, 52sylibd 149 . . . . . . . 8 (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
5429, 53mtod 664 . . . . . . 7 (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
55 pcxqcl 12450 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
56 zq 9691 . . . . . . . . . . 11 ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ)
5756orim1i 761 . . . . . . . . . 10 (((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℤ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
5855, 57syl 14 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
591, 7, 58syl2anc 411 . . . . . . . 8 (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℚ ∨ (𝑃 pCnt (𝐴 + 𝐵)) = +∞))
60 xqltnle 10336 . . . . . . . 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))))
6159, 26, 60syl2anc 411 . . . . . . 7 (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
6254, 61mpbird 167 . . . . . 6 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵))
639, 11, 62xrltled 9865 . . . . 5 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
6463, 34breqtrrd 4057 . . . 4 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
651, 7, 16, 64pcadd 12478 . . 3 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
6643, 40, 41addassd 8042 . . . . 5 (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
6746oveq2d 5934 . . . . 5 (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
6866, 67, 493eqtrd 2230 . . . 4 (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
6968oveq2d 5934 . . 3 (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
7065, 69breqtrd 4055 . 2 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
714, 9, 14, 70xrletrid 9871 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 1364  wcel 2164   class class class wbr 4029  (class class class)co 5918  cc 7870  0cc0 7872   + caddc 7875  +∞cpnf 8051  *cxr 8053   < clt 8054  cle 8055  -cneg 8191  cz 9317  cq 9684  cprime 12245   pCnt cpc 12422
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246  df-pc 12423
This theorem is referenced by: (None)
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