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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrtelqelz | Structured version Visualization version GIF version |
Description: zsqrtelqelz 16091 generalized to positive integer roots. (Contributed by Steven Nguyen, 6-Apr-2023.) |
Ref | Expression |
---|---|
zrtelqelz | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴↑𝑐(1 / 𝑁)) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qdencl 16074 | . . . . 5 ⊢ ((𝐴↑𝑐(1 / 𝑁)) ∈ ℚ → (denom‘(𝐴↑𝑐(1 / 𝑁))) ∈ ℕ) | |
2 | 1 | 3ad2ant3 1130 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘(𝐴↑𝑐(1 / 𝑁))) ∈ ℕ) |
3 | 2 | nnrpd 12423 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘(𝐴↑𝑐(1 / 𝑁))) ∈ ℝ+) |
4 | 1rp 12387 | . . . 4 ⊢ 1 ∈ ℝ+ | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → 1 ∈ ℝ+) |
6 | simp2 1132 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → 𝑁 ∈ ℕ) | |
7 | 6 | nnzd 12080 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → 𝑁 ∈ ℤ) |
8 | 6 | nnne0d 11681 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → 𝑁 ≠ 0) |
9 | 1exp 13455 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) | |
10 | 7, 9 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (1↑𝑁) = 1) |
11 | zcn 11980 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
12 | 11 | 3ad2ant1 1128 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → 𝐴 ∈ ℂ) |
13 | cxproot 25269 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑐(1 / 𝑁))↑𝑁) = 𝐴) | |
14 | 12, 6, 13 | syl2anc 586 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → ((𝐴↑𝑐(1 / 𝑁))↑𝑁) = 𝐴) |
15 | 14 | fveq2d 6667 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘((𝐴↑𝑐(1 / 𝑁))↑𝑁)) = (denom‘𝐴)) |
16 | zq 12348 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
17 | qden1elz 16090 | . . . . . . . 8 ⊢ (𝐴 ∈ ℚ → ((denom‘𝐴) = 1 ↔ 𝐴 ∈ ℤ)) | |
18 | 16, 17 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → ((denom‘𝐴) = 1 ↔ 𝐴 ∈ ℤ)) |
19 | 18 | ibir 270 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (denom‘𝐴) = 1) |
20 | 19 | 3ad2ant1 1128 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘𝐴) = 1) |
21 | 15, 20 | eqtrd 2855 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘((𝐴↑𝑐(1 / 𝑁))↑𝑁)) = 1) |
22 | simp3 1133 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) | |
23 | 6 | nnnn0d 11949 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → 𝑁 ∈ ℕ0) |
24 | denexp 39265 | . . . . 5 ⊢ (((𝐴↑𝑐(1 / 𝑁)) ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘((𝐴↑𝑐(1 / 𝑁))↑𝑁)) = ((denom‘(𝐴↑𝑐(1 / 𝑁)))↑𝑁)) | |
25 | 22, 23, 24 | syl2anc 586 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘((𝐴↑𝑐(1 / 𝑁))↑𝑁)) = ((denom‘(𝐴↑𝑐(1 / 𝑁)))↑𝑁)) |
26 | 10, 21, 25 | 3eqtr2rd 2862 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → ((denom‘(𝐴↑𝑐(1 / 𝑁)))↑𝑁) = (1↑𝑁)) |
27 | 3, 5, 7, 8, 26 | exp11d 39266 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (denom‘(𝐴↑𝑐(1 / 𝑁))) = 1) |
28 | qden1elz 16090 | . . 3 ⊢ ((𝐴↑𝑐(1 / 𝑁)) ∈ ℚ → ((denom‘(𝐴↑𝑐(1 / 𝑁))) = 1 ↔ (𝐴↑𝑐(1 / 𝑁)) ∈ ℤ)) | |
29 | 28 | 3ad2ant3 1130 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → ((denom‘(𝐴↑𝑐(1 / 𝑁))) = 1 ↔ (𝐴↑𝑐(1 / 𝑁)) ∈ ℤ)) |
30 | 27, 29 | mpbid 234 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴↑𝑐(1 / 𝑁)) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 1c1 10531 / cdiv 11290 ℕcn 11631 ℕ0cn0 11891 ℤcz 11975 ℚcq 12342 ℝ+crp 12383 ↑cexp 13426 denomcdenom 16067 ↑𝑐ccxp 25135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 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-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-fi 8868 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-fac 13631 df-bc 13660 df-hash 13688 df-shft 14419 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-limsup 14821 df-clim 14838 df-rlim 14839 df-sum 15036 df-ef 15414 df-sin 15416 df-cos 15417 df-pi 15419 df-dvds 15601 df-gcd 15837 df-numer 16068 df-denom 16069 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-hom 16582 df-cco 16583 df-rest 16689 df-topn 16690 df-0g 16708 df-gsum 16709 df-topgen 16710 df-pt 16711 df-prds 16714 df-xrs 16768 df-qtop 16773 df-imas 16774 df-xps 16776 df-mre 16850 df-mrc 16851 df-acs 16853 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-submnd 17950 df-mulg 18218 df-cntz 18440 df-cmn 18901 df-psmet 20530 df-xmet 20531 df-met 20532 df-bl 20533 df-mopn 20534 df-fbas 20535 df-fg 20536 df-cnfld 20539 df-top 21495 df-topon 21512 df-topsp 21534 df-bases 21547 df-cld 21620 df-ntr 21621 df-cls 21622 df-nei 21699 df-lp 21737 df-perf 21738 df-cn 21828 df-cnp 21829 df-haus 21916 df-tx 22163 df-hmeo 22356 df-fil 22447 df-fm 22539 df-flim 22540 df-flf 22541 df-xms 22923 df-ms 22924 df-tms 22925 df-cncf 23479 df-limc 24459 df-dv 24460 df-log 25136 df-cxp 25137 |
This theorem is referenced by: rtprmirr 39271 |
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