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Theorem logbgcd1irraplemexp 15100
Description: Lemma for logbgcd1irrap 15102. 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 9631 . . . . . . . . . 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 9631 . . . . . . . . . 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 12164 . . . . . . . . 9  |-  ( ( X  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( X  gcd  B
)  =  1  -> 
( ( X ^ N )  gcd  B
)  =  1 ) )
104, 7, 8, 9syl3anc 1249 . . . . . . . 8  |-  ( ph  ->  ( ( X  gcd  B )  =  1  -> 
( ( X ^ N )  gcd  B
)  =  1 ) )
111, 10mpd 13 . . . . . . 7  |-  ( ph  ->  ( ( X ^ N )  gcd  B
)  =  1 )
1211ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  (
( X ^ N
)  gcd  B )  =  1 )
13 1red 8034 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  RR )
14 eluz2gt1 9667 . . . . . . . . . . . . . 14  |-  ( B  e.  ( ZZ>= `  2
)  ->  1  <  B )
155, 14syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  1  <  B )
1613, 15gtned 8132 . . . . . . . . . . . 12  |-  ( ph  ->  B  =/=  1 )
1716neneqd 2385 . . . . . . . . . . 11  |-  ( ph  ->  -.  B  =  1 )
187nnzd 9438 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  ZZ )
19 gcdid 12123 . . . . . . . . . . . . . 14  |-  ( B  e.  ZZ  ->  ( B  gcd  B )  =  ( abs `  B
) )
2018, 19syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  gcd  B
)  =  ( abs `  B ) )
217nnred 8995 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  RR )
227nnnn0d 9293 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  NN0 )
2322nn0ge0d 9296 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <_  B )
2421, 23absidd 11311 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  B
)  =  B )
2520, 24eqtrd 2226 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  gcd  B
)  =  B )
2625eqeq1d 2202 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  gcd  B )  =  1  <->  B  =  1 ) )
2717, 26mtbird 674 . . . . . . . . . 10  |-  ( ph  ->  -.  ( B  gcd  B )  =  1 )
2827adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  NN )  ->  -.  ( B  gcd  B )  =  1 )
2918adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  NN )  ->  B  e.  ZZ )
30 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  M  e.  NN )  ->  M  e.  NN )
31 rpexp 12291 . . . . . . . . . 10  |-  ( ( B  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  (
( ( B ^ M )  gcd  B
)  =  1  <->  ( B  gcd  B )  =  1 ) )
3229, 29, 30, 31syl3anc 1249 . . . . . . . . 9  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( ( B ^ M
)  gcd  B )  =  1  <->  ( B  gcd  B )  =  1 ) )
3328, 32mtbird 674 . . . . . . . 8  |-  ( (
ph  /\  M  e.  NN )  ->  -.  (
( B ^ M
)  gcd  B )  =  1 )
3433adantr 276 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  -.  ( ( B ^ M )  gcd  B
)  =  1 )
35 oveq1 5925 . . . . . . . . . 10  |-  ( ( X ^ N )  =  ( B ^ M )  ->  (
( X ^ N
)  gcd  B )  =  ( ( B ^ M )  gcd 
B ) )
3635eqeq1d 2202 . . . . . . . . 9  |-  ( ( X ^ N )  =  ( B ^ M )  ->  (
( ( X ^ N )  gcd  B
)  =  1  <->  (
( B ^ M
)  gcd  B )  =  1 ) )
3736eqcoms 2196 . . . . . . . 8  |-  ( ( B ^ M )  =  ( X ^ N )  ->  (
( ( X ^ N )  gcd  B
)  =  1  <->  (
( B ^ M
)  gcd  B )  =  1 ) )
3837adantl 277 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  (
( ( X ^ N )  gcd  B
)  =  1  <->  (
( B ^ M
)  gcd  B )  =  1 ) )
3934, 38mtbird 674 . . . . . 6  |-  ( ( ( ph  /\  M  e.  NN )  /\  ( B ^ M )  =  ( X ^ N
) )  ->  -.  ( ( X ^ N )  gcd  B
)  =  1 )
4012, 39pm2.65da 662 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  -.  ( B ^ M )  =  ( X ^ N
) )
4140neqcomd 2198 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  -.  ( X ^ N )  =  ( B ^ M
) )
4241neqned 2371 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( X ^ N )  =/=  ( B ^ M
) )
434nnzd 9438 . . . . . 6  |-  ( ph  ->  X  e.  ZZ )
4443adantr 276 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  X  e.  ZZ )
458nnnn0d 9293 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
4645adantr 276 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  N  e. 
NN0 )
47 zexpcl 10625 . . . . 5  |-  ( ( X  e.  ZZ  /\  N  e.  NN0 )  -> 
( X ^ N
)  e.  ZZ )
4844, 46, 47syl2anc 411 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( X ^ N )  e.  ZZ )
4930nnnn0d 9293 . . . . 5  |-  ( (
ph  /\  M  e.  NN )  ->  M  e. 
NN0 )
50 zexpcl 10625 . . . . 5  |-  ( ( B  e.  ZZ  /\  M  e.  NN0 )  -> 
( B ^ M
)  e.  ZZ )
5129, 49, 50syl2anc 411 . . . 4  |-  ( (
ph  /\  M  e.  NN )  ->  ( B ^ M )  e.  ZZ )
52 zapne 9391 . . . 4  |-  ( ( ( X ^ N
)  e.  ZZ  /\  ( B ^ M )  e.  ZZ )  -> 
( ( X ^ N ) #  ( B ^ M )  <->  ( X ^ N )  =/=  ( B ^ M ) ) )
5348, 51, 52syl2anc 411 . . 3  |-  ( (
ph  /\  M  e.  NN )  ->  ( ( X ^ N ) #  ( B ^ M
)  <->  ( X ^ N )  =/=  ( B ^ M ) ) )
5442, 53mpbird 167 . 2  |-  ( (
ph  /\  M  e.  NN )  ->  ( X ^ N ) #  ( B ^ M ) )
557nnrpd 9760 . . . . . 6  |-  ( ph  ->  B  e.  RR+ )
5655adantr 276 . . . . 5  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B  e.  RR+ )
57 logbgcd1irraplem.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
5857adantr 276 . . . . 5  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  M  e.  ZZ )
5956, 58rpexpcld 10768 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  e.  RR+ )
6059rpred 9762 . . 3  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  e.  RR )
614nnred 8995 . . . . 5  |-  ( ph  ->  X  e.  RR )
6261, 45reexpcld 10761 . . . 4  |-  ( ph  ->  ( X ^ N
)  e.  RR )
6362adantr 276 . . 3  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( X ^ N )  e.  RR )
64 1red 8034 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  e.  RR )
65 1rp 9723 . . . . . . 7  |-  1  e.  RR+
6665a1i 9 . . . . . 6  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  e.  RR+ )
6721adantr 276 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B  e.  RR )
68 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  -u M  e.  NN0 )
697nnge1d 9025 . . . . . . . . 9  |-  ( ph  ->  1  <_  B )
7069adantr 276 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <_  B )
7167, 68, 70expge1d 10763 . . . . . . 7  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <_  ( B ^ -u M
) )
7267recnd 8048 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B  e.  CC )
737nnap0d 9028 . . . . . . . . 9  |-  ( ph  ->  B #  0 )
7473adantr 276 . . . . . . . 8  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  B #  0 )
7572, 74, 58expnegapd 10751 . . . . . . 7  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ -u M )  =  ( 1  / 
( B ^ M
) ) )
7671, 75breqtrd 4055 . . . . . 6  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <_  ( 1  /  ( B ^ M ) ) )
7766, 59, 76lerec2d 9784 . . . . 5  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  <_ 
( 1  /  1
) )
78 1div1e1 8723 . . . . 5  |-  ( 1  /  1 )  =  1
7977, 78breqtrdi 4070 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  <_ 
1 )
80 eluz2gt1 9667 . . . . . . 7  |-  ( X  e.  ( ZZ>= `  2
)  ->  1  <  X )
812, 80syl 14 . . . . . 6  |-  ( ph  ->  1  <  X )
82 expgt1 10648 . . . . . 6  |-  ( ( X  e.  RR  /\  N  e.  NN  /\  1  <  X )  ->  1  <  ( X ^ N
) )
8361, 8, 81, 82syl3anc 1249 . . . . 5  |-  ( ph  ->  1  <  ( X ^ N ) )
8483adantr 276 . . . 4  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  1  <  ( X ^ N
) )
8560, 64, 63, 79, 84lelttrd 8144 . . 3  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( B ^ M )  < 
( X ^ N
) )
8660, 63, 85gtapd 8656 . 2  |-  ( (
ph  /\  -u M  e. 
NN0 )  ->  ( X ^ N ) #  ( B ^ M ) )
87 elznn 9333 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  e.  NN  \/  -u M  e.  NN0 ) ) )
8857, 87sylib 122 . . 3  |-  ( ph  ->  ( M  e.  RR  /\  ( M  e.  NN  \/  -u M  e.  NN0 ) ) )
8988simprd 114 . 2  |-  ( ph  ->  ( M  e.  NN  \/  -u M  e.  NN0 ) )
9054, 86, 89mpjaodan 799 1  |-  ( ph  ->  ( X ^ N
) #  ( B ^ M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2164    =/= wne 2364   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873    < clt 8054    <_ cle 8055   -ucneg 8191   # cap 8600    / cdiv 8691   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   RR+crp 9719   ^cexp 10609   abscabs 11141    gcd cgcd 12079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246
This theorem is referenced by:  logbgcd1irraplemap  15101
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