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Theorem logbgcd1irraplemexp 13382
Description: Lemma for logbgcd1irrap 13384. Apartness of  X ^ N and  B ^ M. (Contributed by Jim Kingdon, 11-Jul-2024.)
Hypotheses
Ref Expression
logbgcd1irraplem.x  |-  ( ph  ->  X  e.  ( ZZ>= ` 
2 ) )
logbgcd1irraplem.b  |-  ( ph  ->  B  e.  ( ZZ>= ` 
2 ) )
logbgcd1irraplem.rp  |-  ( ph  ->  ( X  gcd  B
)  =  1 )
logbgcd1irraplem.m  |-  ( ph  ->  M  e.  ZZ )
logbgcd1irraplem.n  |-  ( ph  ->  N  e.  NN )
Assertion
Ref Expression
logbgcd1irraplemexp  |-  ( ph  ->  ( X ^ N
) #  ( B ^ M ) )

Proof of Theorem logbgcd1irraplemexp
StepHypRef Expression
1 logbgcd1irraplem.rp . . . . . . . 8  |-  ( ph  ->  ( X  gcd  B
)  =  1 )
2 logbgcd1irraplem.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( ZZ>= ` 
2 ) )
3 eluz2nn 9483 . . . . . . . . . 10  |-  ( X  e.  ( ZZ>= `  2
)  ->  X  e.  NN )
42, 3syl 14 . . . . . . . . 9  |-  ( ph  ->  X  e.  NN )
5 logbgcd1irraplem.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( ZZ>= ` 
2 ) )
6 eluz2nn 9483 . . . . . . . . . 10  |-  ( B  e.  ( ZZ>= `  2
)  ->  B  e.  NN )
75, 6syl 14 . . . . . . . . 9  |-  ( ph  ->  B  e.  NN )
8 logbgcd1irraplem.n . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
9 rplpwr 11927 . . . . . . . . 9  |-  ( ( X  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( X  gcd  B
)  =  1  -> 
( ( X ^ N )  gcd  B
)  =  1 ) )
104, 7, 8, 9syl3anc 1220 . . . . . . . 8  |-  ( ph  ->  ( ( X  gcd  B )  =  1  -> 
( ( X ^ N )  gcd  B
)  =  1 ) )
111, 10mpd 13 . . . . . . 7  |-  ( ph  ->  ( ( X ^ N )  gcd  B
)  =  1 )
1211ad2antrr 480 . . . . . 6  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  (
( X ^ N
)  gcd  B )  =  1 )
13 1red 7896 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  RR )
14 eluz2gt1 9519 . . . . . . . . . . . . . 14  |-  ( B  e.  ( ZZ>= `  2
)  ->  1  <  B )
155, 14syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  1  <  B )
1613, 15gtned 7993 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  1 )
1716neneqd 2348 . . . . . . . . . . 11  |-  ( ph  ->  -.  B  =  1 )
187nnzd 9291 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  ZZ )
19 gcdid 11886 . . . . . . . . . . . . . 14  |-  ( B  e.  ZZ  ->  ( B  gcd  B )  =  ( abs `  B
) )
2018, 19syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  gcd  B
)  =  ( abs `  B ) )
217nnred 8852 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  RR )
227nnnn0d 9149 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  NN0 )
2322nn0ge0d 9152 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <_  B )
2421, 23absidd 11079 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  B
)  =  B )
2520, 24eqtrd 2190 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  gcd  B
)  =  B )
2625eqeq1d 2166 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  gcd  B )  =  1  <->  B  =  1 ) )
2717, 26mtbird 663 . . . . . . . . . 10  |-  ( ph  ->  -.  ( B  gcd  B )  =  1 )
2827adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  NN )  ->  -.  ( B  gcd  B )  =  1 )
2918adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  NN )  ->  B  e.  ZZ )
30 simpr 109 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  NN )  ->  M  e.  NN )
31 rpexp 12044 . . . . . . . . . 10  |-  ( ( B  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  (
( ( B ^ M )  gcd  B
)  =  1  <->  ( B  gcd  B )  =  1 ) )
3229, 29, 30, 31syl3anc 1220 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( ( B ^ M
)  gcd  B )  =  1  <->  ( B  gcd  B )  =  1 ) )
3328, 32mtbird 663 . . . . . . . 8  |-  ( (
ph  /\  M  e.  NN )  ->  -.  (
( B ^ M
)  gcd  B )  =  1 )
3433adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  -.  ( ( B ^ M )  gcd  B
)  =  1 )
35 oveq1 5834 . . . . . . . . . 10  |-  ( ( X ^ N )  =  ( B ^ M )  ->  (
( X ^ N
)  gcd  B )  =  ( ( B ^ M )  gcd 
B ) )
3635eqeq1d 2166 . . . . . . . . 9  |-  ( ( X ^ N )  =  ( B ^ M )  ->  (
( ( X ^ N )  gcd  B
)  =  1  <->  (
( B ^ M
)  gcd  B )  =  1 ) )
3736eqcoms 2160 . . . . . . . 8  |-  ( ( B ^ M )  =  ( X ^ N )  ->  (
( ( X ^ N )  gcd  B
)  =  1  <->  (
( B ^ M
)  gcd  B )  =  1 ) )
3837adantl 275 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  (
( ( X ^ N )  gcd  B
)  =  1  <->  (
( B ^ M
)  gcd  B )  =  1 ) )
3934, 38mtbird 663 . . . . . 6  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  -.  ( ( X ^ N )  gcd  B
)  =  1 )
4012, 39pm2.65da 651 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  -.  ( B ^ M )  =  ( X ^ N
) )
4140neqcomd 2162 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  -.  ( X ^ N )  =  ( B ^ M
) )
4241neqned 2334 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( X ^ N )  =/=  ( B ^ M
) )
434nnzd 9291 . . . . . 6  |-  ( ph  ->  X  e.  ZZ )
4443adantr 274 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  X  e.  ZZ )
458nnnn0d 9149 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
4645adantr 274 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  N  e. 
NN0 )
47 zexpcl 10444 . . . . 5  |-  ( ( X  e.  ZZ  /\  N  e.  NN0 )  -> 
( X ^ N
)  e.  ZZ )
4844, 46, 47syl2anc 409 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( X ^ N )  e.  ZZ )
4930nnnn0d 9149 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  M  e. 
NN0 )
50 zexpcl 10444 . . . . 5  |-  ( ( B  e.  ZZ  /\  M  e.  NN0 )  -> 
( B ^ M
)  e.  ZZ )
5129, 49, 50syl2anc 409 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( B ^ M )  e.  ZZ )
52 zapne 9244 . . . 4  |-  ( ( ( X ^ N
)  e.  ZZ  /\  ( B ^ M )  e.  ZZ )  -> 
( ( X ^ N ) #  ( B ^ M )  <->  ( X ^ N )  =/=  ( B ^ M ) ) )
5348, 51, 52syl2anc 409 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( X ^ N ) #  ( B ^ M
)  <->  ( X ^ N )  =/=  ( B ^ M ) ) )
5442, 53mpbird 166 . 2  |-  ( (
ph  /\  M  e.  NN )  ->  ( X ^ N ) #  ( B ^ M ) )
557nnrpd 9608 . . . . . 6  |-  ( ph  ->  B  e.  RR+ )
5655adantr 274 . . . . 5  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B  e.  RR+ )
57 logbgcd1irraplem.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
5857adantr 274 . . . . 5  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  M  e.  ZZ )
5956, 58rpexpcld 10585 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  e.  RR+ )
6059rpred 9610 . . 3  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  e.  RR )
614nnred 8852 . . . . 5  |-  ( ph  ->  X  e.  RR )
6261, 45reexpcld 10578 . . . 4  |-  ( ph  ->  ( X ^ N
)  e.  RR )
6362adantr 274 . . 3  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( X ^ N )  e.  RR )
64 1red 7896 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  e.  RR )
65 1rp 9571 . . . . . . 7  |-  1  e.  RR+
6665a1i 9 . . . . . 6  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  e.  RR+ )
6721adantr 274 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B  e.  RR )
68 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  -u M  e.  NN0 )
697nnge1d 8882 . . . . . . . . 9  |-  ( ph  ->  1  <_  B )
7069adantr 274 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <_  B )
7167, 68, 70expge1d 10580 . . . . . . 7  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <_  ( B ^ -u M
) )
7267recnd 7909 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B  e.  CC )
737nnap0d 8885 . . . . . . . . 9  |-  ( ph  ->  B #  0 )
7473adantr 274 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B #  0 )
7572, 74, 58expnegapd 10568 . . . . . . 7  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ -u M )  =  ( 1  / 
( B ^ M
) ) )
7671, 75breqtrd 3993 . . . . . 6  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <_  ( 1  /  ( B ^ M ) ) )
7766, 59, 76lerec2d 9632 . . . . 5  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  <_ 
( 1  /  1
) )
78 1div1e1 8582 . . . . 5  |-  ( 1  /  1 )  =  1
7977, 78breqtrdi 4008 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  <_ 
1 )
80 eluz2gt1 9519 . . . . . . 7  |-  ( X  e.  ( ZZ>= `  2
)  ->  1  <  X )
812, 80syl 14 . . . . . 6  |-  ( ph  ->  1  <  X )
82 expgt1 10467 . . . . . 6  |-  ( ( X  e.  RR  /\  N  e.  NN  /\  1  <  X )  ->  1  <  ( X ^ N
) )
8361, 8, 81, 82syl3anc 1220 . . . . 5  |-  ( ph  ->  1  <  ( X ^ N ) )
8483adantr 274 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <  ( X ^ N
) )
8560, 64, 63, 79, 84lelttrd 8005 . . 3  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  < 
( X ^ N
) )
8660, 63, 85gtapd 8517 . 2  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( X ^ N ) #  ( B ^ M ) )
87 elznn 9189 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  e.  NN  \/  -u M  e.  NN0 ) ) )
8857, 87sylib 121 . . 3  |-  ( ph  ->  ( M  e.  RR  /\  ( M  e.  NN  \/  -u M  e.  NN0 ) ) )
8988simprd 113 . 2  |-  ( ph  ->  ( M  e.  NN  \/  -u M  e.  NN0 ) )
9054, 86, 89mpjaodan 788 1  |-  ( ph  ->  ( X ^ N
) #  ( B ^ M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1335    e. wcel 2128    =/= wne 2327   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   RRcr 7734   0cc0 7735   1c1 7736    < clt 7915    <_ cle 7916   -ucneg 8052   # cap 8461    / cdiv 8550   NNcn 8839   2c2 8890   NN0cn0 9096   ZZcz 9173   ZZ>=cuz 9445   RR+crp 9567   ^cexp 10428   abscabs 10909    gcd cgcd 11842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-1o 6366  df-2o 6367  df-er 6483  df-en 6689  df-sup 6931  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696  df-gcd 11843  df-prm 12001
This theorem is referenced by:  logbgcd1irraplemap  13383
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