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Theorem pcadd2 15518
Description: The inequality of pcadd 15517 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 pcadd2.3 . . 3 (𝜑𝐵 ∈ ℚ)
4 pcadd2.4 . . . 4 (𝜑 → (𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵))
5 pcxcl 15489 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ*)
61, 2, 5syl2anc 692 . . . . 5 (𝜑 → (𝑃 pCnt 𝐴) ∈ ℝ*)
7 pcxcl 15489 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈ ℝ*)
81, 3, 7syl2anc 692 . . . . 5 (𝜑 → (𝑃 pCnt 𝐵) ∈ ℝ*)
9 xrltle 11926 . . . . 5 (((𝑃 pCnt 𝐴) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)))
106, 8, 9syl2anc 692 . . . 4 (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)))
114, 10mpd 15 . . 3 (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
121, 2, 3, 11pcadd 15517 . 2 (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
13 qaddcl 11748 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
142, 3, 13syl2anc 692 . . . 4 (𝜑 → (𝐴 + 𝐵) ∈ ℚ)
15 qnegcl 11749 . . . . 5 (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
163, 15syl 17 . . . 4 (𝜑 → -𝐵 ∈ ℚ)
17 xrltnle 10049 . . . . . . . . . 10 (((𝑃 pCnt 𝐴) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
186, 8, 17syl2anc 692 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt 𝐴) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
194, 18mpbid 222 . . . . . . . 8 (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
201adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → 𝑃 ∈ ℙ)
2116adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → -𝐵 ∈ ℚ)
2214adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝐴 + 𝐵) ∈ ℚ)
23 pcneg 15502 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
241, 3, 23syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝑃 pCnt -𝐵) = (𝑃 pCnt 𝐵))
2524breq1d 4623 . . . . . . . . . . . 12 (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
2625biimpar 502 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
2720, 21, 22, 26pcadd 15517 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))))
2827ex 450 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵)))))
29 qcn 11746 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
303, 29syl 17 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ ℂ)
3130negcld 10323 . . . . . . . . . . . . 13 (𝜑 → -𝐵 ∈ ℂ)
32 qcn 11746 . . . . . . . . . . . . . 14 (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
332, 32syl 17 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
3431, 33, 30add12d 10206 . . . . . . . . . . . 12 (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = (𝐴 + (-𝐵 + 𝐵)))
3531, 30addcomd 10182 . . . . . . . . . . . . . 14 (𝜑 → (-𝐵 + 𝐵) = (𝐵 + -𝐵))
3630negidd 10326 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 + -𝐵) = 0)
3735, 36eqtrd 2655 . . . . . . . . . . . . 13 (𝜑 → (-𝐵 + 𝐵) = 0)
3837oveq2d 6620 . . . . . . . . . . . 12 (𝜑 → (𝐴 + (-𝐵 + 𝐵)) = (𝐴 + 0))
3933addid1d 10180 . . . . . . . . . . . 12 (𝜑 → (𝐴 + 0) = 𝐴)
4034, 38, 393eqtrd 2659 . . . . . . . . . . 11 (𝜑 → (-𝐵 + (𝐴 + 𝐵)) = 𝐴)
4140oveq2d 6620 . . . . . . . . . 10 (𝜑 → (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) = (𝑃 pCnt 𝐴))
4224, 41breq12d 4626 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt -𝐵) ≤ (𝑃 pCnt (-𝐵 + (𝐴 + 𝐵))) ↔ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
4328, 42sylibd 229 . . . . . . . 8 (𝜑 → ((𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
4419, 43mtod 189 . . . . . . 7 (𝜑 → ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
45 pcxcl 15489 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ (𝐴 + 𝐵) ∈ ℚ) → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ*)
461, 14, 45syl2anc 692 . . . . . . . 8 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ*)
47 xrltnle 10049 . . . . . . . 8 (((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
4846, 8, 47syl2anc 692 . . . . . . 7 (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) ↔ ¬ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
4944, 48mpbird 247 . . . . . 6 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵))
50 xrltle 11926 . . . . . . 7 (((𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵)))
5146, 8, 50syl2anc 692 . . . . . 6 (𝜑 → ((𝑃 pCnt (𝐴 + 𝐵)) < (𝑃 pCnt 𝐵) → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵)))
5249, 51mpd 15 . . . . 5 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐵))
5352, 24breqtrrd 4641 . . . 4 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt -𝐵))
541, 14, 16, 53pcadd 15517 . . 3 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)))
5533, 30, 31addassd 10006 . . . . 5 (𝜑 → ((𝐴 + 𝐵) + -𝐵) = (𝐴 + (𝐵 + -𝐵)))
5636oveq2d 6620 . . . . 5 (𝜑 → (𝐴 + (𝐵 + -𝐵)) = (𝐴 + 0))
5755, 56, 393eqtrd 2659 . . . 4 (𝜑 → ((𝐴 + 𝐵) + -𝐵) = 𝐴)
5857oveq2d 6620 . . 3 (𝜑 → (𝑃 pCnt ((𝐴 + 𝐵) + -𝐵)) = (𝑃 pCnt 𝐴))
5954, 58breqtrd 4639 . 2 (𝜑 → (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))
60 xrletri3 11929 . . 3 (((𝑃 pCnt 𝐴) ∈ ℝ* ∧ (𝑃 pCnt (𝐴 + 𝐵)) ∈ ℝ*) → ((𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)) ↔ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ∧ (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))))
616, 46, 60syl2anc 692 . 2 (𝜑 → ((𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)) ↔ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)) ∧ (𝑃 pCnt (𝐴 + 𝐵)) ≤ (𝑃 pCnt 𝐴))))
6212, 59, 61mpbir2and 956 1 (𝜑 → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝐴 + 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987   class class class wbr 4613  (class class class)co 6604  cc 9878  0cc0 9880   + caddc 9883  *cxr 10017   < clt 10018  cle 10019  -cneg 10211  cq 11732  cprime 15309   pCnt cpc 15465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-dvds 14908  df-gcd 15141  df-prm 15310  df-pc 15466
This theorem is referenced by:  sylow1lem1  17934
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