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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 6557 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)) | |
2 | 1 | ralrimivw 2950 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)) |
3 | nn0z 11277 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
4 | nn0z 11277 | . . . 4 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ) | |
5 | zq 11670 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
6 | pcxcl 15403 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑝 pCnt 𝐴) ∈ ℝ*) | |
7 | 5, 6 | sylan2 490 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑝 pCnt 𝐴) ∈ ℝ*) |
8 | zq 11670 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℚ) | |
9 | pcxcl 15403 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑝 pCnt 𝐵) ∈ ℝ*) | |
10 | 8, 9 | sylan2 490 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (𝑝 pCnt 𝐵) ∈ ℝ*) |
11 | 7, 10 | anim12dan 878 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)) → ((𝑝 pCnt 𝐴) ∈ ℝ* ∧ (𝑝 pCnt 𝐵) ∈ ℝ*)) |
12 | xrletri3 11861 | . . . . . . . . 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 468 | . . . . . . 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 2968 | . . . . . 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 3046 | . . . . . 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 | syl6bb 275 | . . . . 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 15421 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵))) | |
19 | pc2dvds 15421 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴))) | |
20 | 19 | ancoms 468 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ (𝑝 pCnt 𝐴))) |
21 | 18, 20 | anbi12d 743 | . . . . 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 270 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴))) |
23 | 3, 4, 22 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴))) |
24 | dvdseq 14874 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴)) → 𝐴 = 𝐵) | |
25 | 24 | ex 449 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴) → 𝐴 = 𝐵)) |
26 | 23, 25 | sylbid 229 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) → 𝐴 = 𝐵)) |
27 | 2, 26 | impbid2 215 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 (class class class)co 6549 ℝ*cxr 9952 ≤ cle 9954 ℕ0cn0 11169 ℤcz 11254 ℚcq 11664 ∥ cdvds 14821 ℙcprime 15223 pCnt cpc 15379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 |
This theorem is referenced by: pcprod 15437 prmreclem2 15459 1arith 15469 isppw2 24641 sqf11 24665 bposlem3 24811 |
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