![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pcrec | Structured version Visualization version GIF version |
Description: Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.) |
Ref | Expression |
---|---|
pcrec | ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (1 / 𝐴)) = -(𝑃 pCnt 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12492 | . . . . . 6 ⊢ 1 ∈ ℤ | |
2 | zq 12834 | . . . . . 6 ⊢ (1 ∈ ℤ → 1 ∈ ℚ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1 ∈ ℚ |
4 | ax-1ne0 11079 | . . . . 5 ⊢ 1 ≠ 0 | |
5 | 3, 4 | pm3.2i 472 | . . . 4 ⊢ (1 ∈ ℚ ∧ 1 ≠ 0) |
6 | pcqdiv 16689 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (1 ∈ ℚ ∧ 1 ≠ 0) ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (1 / 𝐴)) = ((𝑃 pCnt 1) − (𝑃 pCnt 𝐴))) | |
7 | 5, 6 | mp3an2 1450 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (1 / 𝐴)) = ((𝑃 pCnt 1) − (𝑃 pCnt 𝐴))) |
8 | pc1 16687 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0) | |
9 | 8 | adantr 482 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 1) = 0) |
10 | 9 | oveq1d 7367 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → ((𝑃 pCnt 1) − (𝑃 pCnt 𝐴)) = (0 − (𝑃 pCnt 𝐴))) |
11 | 7, 10 | eqtrd 2778 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (1 / 𝐴)) = (0 − (𝑃 pCnt 𝐴))) |
12 | df-neg 11347 | . 2 ⊢ -(𝑃 pCnt 𝐴) = (0 − (𝑃 pCnt 𝐴)) | |
13 | 11, 12 | eqtr4di 2796 | 1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (1 / 𝐴)) = -(𝑃 pCnt 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 (class class class)co 7352 0cc0 11010 1c1 11011 − cmin 11344 -cneg 11345 / cdiv 11771 ℤcz 12458 ℚcq 12828 ℙcprime 16507 pCnt cpc 16668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-inf 9338 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-n0 12373 df-z 12459 df-uz 12723 df-q 12829 df-rp 12871 df-fl 13652 df-mod 13730 df-seq 13862 df-exp 13923 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-dvds 16097 df-gcd 16335 df-prm 16508 df-pc 16669 |
This theorem is referenced by: pcexp 16691 |
Copyright terms: Public domain | W3C validator |