Theorem logbgcd1irraplemexp 13526

Description: Lemma for logbgcd1irrap 13528. 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

Step	Hyp	Ref	Expression
1		logbgcd1irraplem.rp	. . . . . . . 8 |- ( ph -> ( X gcd B ) = 1 )
2		logbgcd1irraplem.x	. . . . . . . . . 10 |- ( ph -> X e. ( ZZ>= ` 2 ) )
3		eluz2nn 9504	. . . . . . . . . 10 |- ( X e. ( ZZ>= ` 2 ) -> X e. NN )
4	2, 3	syl 14	. . . . . . . . 9 |- ( ph -> X e. NN )
5		logbgcd1irraplem.b	. . . . . . . . . 10 |- ( ph -> B e. ( ZZ>= ` 2 ) )
6		eluz2nn 9504	. . . . . . . . . 10 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
7	5, 6	syl 14	. . . . . . . . 9 |- ( ph -> B e. NN )
8		logbgcd1irraplem.n	. . . . . . . . 9 |- ( ph -> N e. NN )
9		rplpwr 11960	. . . . . . . . 9 |- ( ( X e. NN /\ B e. NN /\ N e. NN ) -> ( ( X gcd B ) = 1 -> ( ( X ^ N ) gcd B ) = 1 ) )
10	4, 7, 8, 9	syl3anc 1228	. . . . . . . 8 |- ( ph -> ( ( X gcd B ) = 1 -> ( ( X ^ N ) gcd B ) = 1 ) )
11	1, 10	mpd 13	. . . . . . 7 |- ( ph -> ( ( X ^ N ) gcd B ) = 1 )
12	11	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> ( ( X ^ N ) gcd B ) = 1 )
13		1red 7914	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR )
14		eluz2gt1 9540	. . . . . . . . . . . . . 14 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
15	5, 14	syl 14	. . . . . . . . . . . . 13 |- ( ph -> 1 < B )
16	13, 15	gtned 8011	. . . . . . . . . . . 12 |- ( ph -> B =/= 1 )
17	16	neneqd 2357	. . . . . . . . . . 11 |- ( ph -> -. B = 1 )
18	7	nnzd 9312	. . . . . . . . . . . . . 14 |- ( ph -> B e. ZZ )
19		gcdid 11919	. . . . . . . . . . . . . 14 |- ( B e. ZZ -> ( B gcd B ) = ( abs ` B ) )
20	18, 19	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( B gcd B ) = ( abs ` B ) )
21	7	nnred 8870	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
22	7	nnnn0d 9167	. . . . . . . . . . . . . . 15 |- ( ph -> B e. NN0 )
23	22	nn0ge0d 9170	. . . . . . . . . . . . . 14 |- ( ph -> 0 <_ B )
24	21, 23	absidd 11109	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` B ) = B )
25	20, 24	eqtrd 2198	. . . . . . . . . . . 12 |- ( ph -> ( B gcd B ) = B )
26	25	eqeq1d 2174	. . . . . . . . . . 11 |- ( ph -> ( ( B gcd B ) = 1 <-> B = 1 ) )
27	17, 26	mtbird 663	. . . . . . . . . 10 |- ( ph -> -. ( B gcd B ) = 1 )
28	27	adantr 274	. . . . . . . . 9 |- ( ( ph /\ M e. NN ) -> -. ( B gcd B ) = 1 )
29	18	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ M e. NN ) -> B e. ZZ )
30		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ M e. NN ) -> M e. NN )
31		rpexp 12085	. . . . . . . . . 10 |- ( ( B e. ZZ /\ B e. ZZ /\ M e. NN ) -> ( ( ( B ^ M ) gcd B ) = 1 <-> ( B gcd B ) = 1 ) )
32	29, 29, 30, 31	syl3anc 1228	. . . . . . . . 9 |- ( ( ph /\ M e. NN ) -> ( ( ( B ^ M ) gcd B ) = 1 <-> ( B gcd B ) = 1 ) )
33	28, 32	mtbird 663	. . . . . . . 8 |- ( ( ph /\ M e. NN ) -> -. ( ( B ^ M ) gcd B ) = 1 )
34	33	adantr 274	. . . . . . 7 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> -. ( ( B ^ M ) gcd B ) = 1 )
35		oveq1 5849	. . . . . . . . . 10 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( X ^ N ) gcd B ) = ( ( B ^ M ) gcd B ) )
36	35	eqeq1d 2174	. . . . . . . . 9 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
37	36	eqcoms 2168	. . . . . . . 8 |- ( ( B ^ M ) = ( X ^ N ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
38	37	adantl 275	. . . . . . 7 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
39	34, 38	mtbird 663	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> -. ( ( X ^ N ) gcd B ) = 1 )
40	12, 39	pm2.65da 651	. . . . 5 |- ( ( ph /\ M e. NN ) -> -. ( B ^ M ) = ( X ^ N ) )
41	40	neqcomd 2170	. . . 4 |- ( ( ph /\ M e. NN ) -> -. ( X ^ N ) = ( B ^ M ) )
42	41	neqned 2343	. . 3 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) =/= ( B ^ M ) )
43	4	nnzd 9312	. . . . . 6 |- ( ph -> X e. ZZ )
44	43	adantr 274	. . . . 5 |- ( ( ph /\ M e. NN ) -> X e. ZZ )
45	8	nnnn0d 9167	. . . . . 6 |- ( ph -> N e. NN0 )
46	45	adantr 274	. . . . 5 |- ( ( ph /\ M e. NN ) -> N e. NN0 )
47		zexpcl 10470	. . . . 5 |- ( ( X e. ZZ /\ N e. NN0 ) -> ( X ^ N ) e. ZZ )
48	44, 46, 47	syl2anc 409	. . . 4 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) e. ZZ )
49	30	nnnn0d 9167	. . . . 5 |- ( ( ph /\ M e. NN ) -> M e. NN0 )
50		zexpcl 10470	. . . . 5 |- ( ( B e. ZZ /\ M e. NN0 ) -> ( B ^ M ) e. ZZ )
51	29, 49, 50	syl2anc 409	. . . 4 |- ( ( ph /\ M e. NN ) -> ( B ^ M ) e. ZZ )
52		zapne 9265	. . . 4 |- ( ( ( X ^ N ) e. ZZ /\ ( B ^ M ) e. ZZ ) -> ( ( X ^ N ) # ( B ^ M ) <-> ( X ^ N ) =/= ( B ^ M ) ) )
53	48, 51, 52	syl2anc 409	. . 3 |- ( ( ph /\ M e. NN ) -> ( ( X ^ N ) # ( B ^ M ) <-> ( X ^ N ) =/= ( B ^ M ) ) )
54	42, 53	mpbird 166	. 2 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) # ( B ^ M ) )
55	7	nnrpd 9630	. . . . . 6 |- ( ph -> B e. RR+ )
56	55	adantr 274	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> B e. RR+ )
57		logbgcd1irraplem.m	. . . . . 6 |- ( ph -> M e. ZZ )
58	57	adantr 274	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> M e. ZZ )
59	56, 58	rpexpcld 10612	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR+ )
60	59	rpred 9632	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR )
61	4	nnred 8870	. . . . 5 |- ( ph -> X e. RR )
62	61, 45	reexpcld 10605	. . . 4 |- ( ph -> ( X ^ N ) e. RR )
63	62	adantr 274	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) e. RR )
64		1red 7914	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR )
65		1rp 9593	. . . . . . 7 |- 1 e. RR+
66	65	a1i 9	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR+ )
67	21	adantr 274	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. RR )
68		simpr 109	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> -u M e. NN0 )
69	7	nnge1d 8900	. . . . . . . . 9 |- ( ph -> 1 <_ B )
70	69	adantr 274	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ B )
71	67, 68, 70	expge1d 10607	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( B ^ -u M ) )
72	67	recnd 7927	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. CC )
73	7	nnap0d 8903	. . . . . . . . 9 |- ( ph -> B # 0 )
74	73	adantr 274	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B # 0 )
75	72, 74, 58	expnegapd 10595	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ -u M ) = ( 1 / ( B ^ M ) ) )
76	71, 75	breqtrd 4008	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( 1 / ( B ^ M ) ) )
77	66, 59, 76	lerec2d 9654	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ ( 1 / 1 ) )
78		1div1e1 8600	. . . . 5 |- ( 1 / 1 ) = 1
79	77, 78	breqtrdi 4023	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ 1 )
80		eluz2gt1 9540	. . . . . . 7 |- ( X e. ( ZZ>= ` 2 ) -> 1 < X )
81	2, 80	syl 14	. . . . . 6 |- ( ph -> 1 < X )
82		expgt1 10493	. . . . . 6 |- ( ( X e. RR /\ N e. NN /\ 1 < X ) -> 1 < ( X ^ N ) )
83	61, 8, 81, 82	syl3anc 1228	. . . . 5 |- ( ph -> 1 < ( X ^ N ) )
84	83	adantr 274	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 < ( X ^ N ) )
85	60, 64, 63, 79, 84	lelttrd 8023	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) < ( X ^ N ) )
86	60, 63, 85	gtapd 8535	. 2 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) # ( B ^ M ) )
87		elznn 9207	. . . 4 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN \/ -u M e. NN0 ) ) )
88	57, 87	sylib 121	. . 3 |- ( ph -> ( M e. RR /\ ( M e. NN \/ -u M e. NN0 ) ) )
89	88	simprd 113	. 2 |- ( ph -> ( M e. NN \/ -u M e. NN0 ) )
90	54, 86, 89	mpjaodan 788	1 |- ( ph -> ( X ^ N ) # ( B ^ M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   < clt 7933   <_ cle 7934   -ucneg 8070   # cap 8479   / cdiv 8568   NNcn 8857   2c2 8908   NN0cn0 9114   ZZcz 9191   ZZ>=cuz 9466   RR+crp 9589   ^cexp 10454   abscabs 10939   gcd cgcd 11875

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876  df-prm 12040

This theorem is referenced by:  logbgcd1irraplemap  13527