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Mirrors > Home > MPE Home > Th. List > pc11 | Structured version Visualization version GIF version |
Description: The prime count function, viewed as a function from ℕ to (ℕ ↑m ℙ), is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
pc11 | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7432 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)) | |
2 | 1 | ralrimivw 3140 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)) |
3 | nn0z 12635 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
4 | nn0z 12635 | . . . 4 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ) | |
5 | zq 12990 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
6 | pcxcl 16863 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑝 pCnt 𝐴) ∈ ℝ*) | |
7 | 5, 6 | sylan2 591 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑝 pCnt 𝐴) ∈ ℝ*) |
8 | zq 12990 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℚ) | |
9 | pcxcl 16863 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑝 pCnt 𝐵) ∈ ℝ*) | |
10 | 8, 9 | sylan2 591 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (𝑝 pCnt 𝐵) ∈ ℝ*) |
11 | 7, 10 | anim12dan 617 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)) → ((𝑝 pCnt 𝐴) ∈ ℝ* ∧ (𝑝 pCnt 𝐵) ∈ ℝ*)) |
12 | xrletri3 13187 | . . . . . . . . 9 ⊢ (((𝑝 pCnt 𝐴) ∈ ℝ* ∧ (𝑝 pCnt 𝐵) ∈ ℝ*) → ((𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ ((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)))) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)) → ((𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ ((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)))) |
14 | 13 | ancoms 457 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ ((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)))) |
15 | 14 | ralbidva 3166 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ ∀𝑝 ∈ ℙ ((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)))) |
16 | r19.26 3101 | . . . . . 6 ⊢ (∀𝑝 ∈ ℙ ((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)) ↔ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴))) | |
17 | 15, 16 | bitrdi 286 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)))) |
18 | pc2dvds 16881 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵))) | |
19 | pc2dvds 16881 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴))) | |
20 | 19 | ancoms 457 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴))) |
21 | 18, 20 | anbi12d 630 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴) ↔ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵) ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴)))) |
22 | 17, 21 | bitr4d 281 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴))) |
23 | 3, 4, 22 | syl2an 594 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴))) |
24 | dvdseq 16316 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴)) → 𝐴 = 𝐵) | |
25 | 24 | ex 411 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴) → 𝐴 = 𝐵)) |
26 | 23, 25 | sylbid 239 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) → 𝐴 = 𝐵)) |
27 | 2, 26 | impbid2 225 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 class class class wbr 5153 (class class class)co 7424 ℝ*cxr 11297 ≤ cle 11299 ℕ0cn0 12524 ℤcz 12610 ℚcq 12984 ∥ cdvds 16256 ℙcprime 16672 pCnt cpc 16838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-fz 13539 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-gcd 16495 df-prm 16673 df-pc 16839 |
This theorem is referenced by: pcprod 16897 prmreclem2 16919 1arith 16929 isppw2 27143 sqf11 27167 bposlem3 27315 aks6d1c2p2 41817 aks6d1c7 41882 |
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