![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > prmexpb | Structured version Visualization version GIF version |
Description: Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Ref | Expression |
---|---|
prmexpb | ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (𝑄↑𝑁) ↔ (𝑃 = 𝑄 ∧ 𝑀 = 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmz 15878 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
2 | 1 | adantr 473 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → 𝑃 ∈ ℤ) |
3 | 2 | 3ad2ant1 1113 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∈ ℤ) |
4 | simp2l 1179 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑀 ∈ ℕ) | |
5 | iddvdsexp 15496 | . . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑃 ∥ (𝑃↑𝑀)) | |
6 | 3, 4, 5 | syl2anc 576 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∥ (𝑃↑𝑀)) |
7 | breq2 4934 | . . . . . . 7 ⊢ ((𝑃↑𝑀) = (𝑄↑𝑁) → (𝑃 ∥ (𝑃↑𝑀) ↔ 𝑃 ∥ (𝑄↑𝑁))) | |
8 | 7 | 3ad2ant3 1115 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 ∥ (𝑃↑𝑀) ↔ 𝑃 ∥ (𝑄↑𝑁))) |
9 | simp1l 1177 | . . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∈ ℙ) | |
10 | simp1r 1178 | . . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑄 ∈ ℙ) | |
11 | simp2r 1180 | . . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑁 ∈ ℕ) | |
12 | prmdvdsexpb 15919 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (𝑄↑𝑁) ↔ 𝑃 = 𝑄)) | |
13 | 9, 10, 11, 12 | syl3anc 1351 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 ∥ (𝑄↑𝑁) ↔ 𝑃 = 𝑄)) |
14 | 8, 13 | bitrd 271 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 ∥ (𝑃↑𝑀) ↔ 𝑃 = 𝑄)) |
15 | 6, 14 | mpbid 224 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 = 𝑄) |
16 | 3 | zred 11903 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∈ ℝ) |
17 | 4 | nnzd 11902 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑀 ∈ ℤ) |
18 | 11 | nnzd 11902 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑁 ∈ ℤ) |
19 | prmgt1 15900 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
20 | 19 | ad2antrr 713 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → 1 < 𝑃) |
21 | 20 | 3adant3 1112 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 1 < 𝑃) |
22 | simp3 1118 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃↑𝑀) = (𝑄↑𝑁)) | |
23 | 15 | oveq1d 6993 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃↑𝑁) = (𝑄↑𝑁)) |
24 | 22, 23 | eqtr4d 2817 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃↑𝑀) = (𝑃↑𝑁)) |
25 | 16, 17, 18, 21, 24 | expcand 13434 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑀 = 𝑁) |
26 | 15, 25 | jca 504 | . . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 = 𝑄 ∧ 𝑀 = 𝑁)) |
27 | 26 | 3expia 1101 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (𝑄↑𝑁) → (𝑃 = 𝑄 ∧ 𝑀 = 𝑁))) |
28 | oveq12 6987 | . 2 ⊢ ((𝑃 = 𝑄 ∧ 𝑀 = 𝑁) → (𝑃↑𝑀) = (𝑄↑𝑁)) | |
29 | 27, 28 | impbid1 217 | 1 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (𝑄↑𝑁) ↔ (𝑃 = 𝑄 ∧ 𝑀 = 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 class class class wbr 4930 (class class class)co 6978 1c1 10338 < clt 10476 ℕcn 11441 ℤcz 11796 ↑cexp 13247 ∥ cdvds 15470 ℙcprime 15874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-sup 8703 df-inf 8704 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-dvds 15471 df-gcd 15707 df-prm 15875 |
This theorem is referenced by: fsumvma 25494 |
Copyright terms: Public domain | W3C validator |