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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 16007 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
2 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → 𝑃 ∈ ℤ) |
3 | 2 | 3ad2ant1 1125 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∈ ℤ) |
4 | simp2l 1191 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑀 ∈ ℕ) | |
5 | iddvdsexp 15621 | . . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑃 ∥ (𝑃↑𝑀)) | |
6 | 3, 4, 5 | syl2anc 584 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∥ (𝑃↑𝑀)) |
7 | breq2 5061 | . . . . . . 7 ⊢ ((𝑃↑𝑀) = (𝑄↑𝑁) → (𝑃 ∥ (𝑃↑𝑀) ↔ 𝑃 ∥ (𝑄↑𝑁))) | |
8 | 7 | 3ad2ant3 1127 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 ∥ (𝑃↑𝑀) ↔ 𝑃 ∥ (𝑄↑𝑁))) |
9 | simp1l 1189 | . . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∈ ℙ) | |
10 | simp1r 1190 | . . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑄 ∈ ℙ) | |
11 | simp2r 1192 | . . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑁 ∈ ℕ) | |
12 | prmdvdsexpb 16048 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (𝑄↑𝑁) ↔ 𝑃 = 𝑄)) | |
13 | 9, 10, 11, 12 | syl3anc 1363 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 ∥ (𝑄↑𝑁) ↔ 𝑃 = 𝑄)) |
14 | 8, 13 | bitrd 280 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 ∥ (𝑃↑𝑀) ↔ 𝑃 = 𝑄)) |
15 | 6, 14 | mpbid 233 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 = 𝑄) |
16 | 3 | zred 12075 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑃 ∈ ℝ) |
17 | 4 | nnzd 12074 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑀 ∈ ℤ) |
18 | 11 | nnzd 12074 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑁 ∈ ℤ) |
19 | prmgt1 16029 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) | |
20 | 19 | ad2antrr 722 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → 1 < 𝑃) |
21 | 20 | 3adant3 1124 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 1 < 𝑃) |
22 | simp3 1130 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃↑𝑀) = (𝑄↑𝑁)) | |
23 | 15 | oveq1d 7160 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃↑𝑁) = (𝑄↑𝑁)) |
24 | 22, 23 | eqtr4d 2856 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃↑𝑀) = (𝑃↑𝑁)) |
25 | 16, 17, 18, 21, 24 | expcand 13604 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → 𝑀 = 𝑁) |
26 | 15, 25 | jca 512 | . . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = (𝑄↑𝑁)) → (𝑃 = 𝑄 ∧ 𝑀 = 𝑁)) |
27 | 26 | 3expia 1113 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (𝑄↑𝑁) → (𝑃 = 𝑄 ∧ 𝑀 = 𝑁))) |
28 | oveq12 7154 | . 2 ⊢ ((𝑃 = 𝑄 ∧ 𝑀 = 𝑁) → (𝑃↑𝑀) = (𝑄↑𝑁)) | |
29 | 27, 28 | impbid1 226 | 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 1c1 10526 < clt 10663 ℕcn 11626 ℤcz 11969 ↑cexp 13417 ∥ cdvds 15595 ℙcprime 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-prm 16004 |
This theorem is referenced by: fsumvma 25716 |
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