Theorem logbgcd1irrap 15961

Description: The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, ( 2 log_b 9 ) # Q where Q is rational. (Contributed by AV, 29-Dec-2022.)

Assertion

Ref	Expression
logbgcd1irrap	|- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( B logb X ) # Q )

Proof of Theorem logbgcd1irrap

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 533	. . 3 |- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> Q e. QQ )
2		elq 9972	. . 3 |- ( Q e. QQ <-> E. m e. ZZ E. n e. NN Q = ( m / n ) )
3	1, 2	sylib 122	. 2 |- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> E. m e. ZZ E. n e. NN Q = ( m / n ) )
4		simp-4l 543	. . . . . 6 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> X e. ( ZZ>= ` 2 ) )
5		simp-4r 544	. . . . . 6 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> B e. ( ZZ>= ` 2 ) )
6		simprl 531	. . . . . . 7 |- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( X gcd B ) = 1 )
7	6	ad2antrr 488	. . . . . 6 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> ( X gcd B ) = 1 )
8		simplrl 537	. . . . . 6 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> m e. ZZ )
9		simplrr 538	. . . . . 6 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> n e. NN )
10	4, 5, 7, 8, 9	logbgcd1irraplemap 15960	. . . . 5 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> ( B logb X ) # ( m / n ) )
11		simpr 110	. . . . 5 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> Q = ( m / n ) )
12	10, 11	breqtrrd 4142	. . . 4 |- ( ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) /\ Q = ( m / n ) ) -> ( B logb X ) # Q )
13	12	ex 115	. . 3 |- ( ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) /\ ( m e. ZZ /\ n e. NN ) ) -> ( Q = ( m / n ) -> ( B logb X ) # Q ) )
14	13	rexlimdvva 2670	. 2 |- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( E. m e. ZZ E. n e. NN Q = ( m / n ) -> ( B logb X ) # Q ) )
15	3, 14	mpd 13	1 |- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( B logb X ) # Q )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   1c1 8144   # cap 8872   / cdiv 8963   NNcn 9254   2c2 9305   ZZcz 9594   ZZ>=cuz 9871   QQcq 9969   gcd cgcd 12674   log_b clogb 15934

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-pre-suploc 8264  ax-addf 8265  ax-mulf 8266

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-ioo 10244  df-ico 10246  df-icc 10247  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-shft 11525  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359  df-e 12360  df-dvds 12499  df-gcd 12675  df-prm 12830  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-cn 15179  df-cnp 15180  df-tx 15244  df-cncf 15562  df-limced 15647  df-dvap 15648  df-relog 15849  df-rpcxp 15850  df-logb 15935

This theorem is referenced by:  2logb9irrap  15968