Theorem logbgcd1irraplemexp 16070

Description: Lemma for logbgcd1irrap 16072. 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 9966	. . . . . . . . . 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 9966	. . . . . . . . . 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 12804	. . . . . . . . 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 1278	. . . . . . . 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 492	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> ( ( X ^ N ) gcd B ) = 1 )
13		1red 8341	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR )
14		eluz2gt1 10002	. . . . . . . . . . . . . 14 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
15	5, 14	syl 14	. . . . . . . . . . . . 13 |- ( ph -> 1 < B )
16	13, 15	gtned 8438	. . . . . . . . . . . 12 |- ( ph -> B =/= 1 )
17	16	neneqd 2441	. . . . . . . . . . 11 |- ( ph -> -. B = 1 )
18	7	nnzd 9767	. . . . . . . . . . . . . 14 |- ( ph -> B e. ZZ )
19		gcdid 12763	. . . . . . . . . . . . . 14 |- ( B e. ZZ -> ( B gcd B ) = ( abs ` B ) )
20	18, 19	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( B gcd B ) = ( abs ` B ) )
21	7	nnred 9317	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
22	7	nnnn0d 9620	. . . . . . . . . . . . . . 15 |- ( ph -> B e. NN0 )
23	22	nn0ge0d 9623	. . . . . . . . . . . . . 14 |- ( ph -> 0 <_ B )
24	21, 23	absidd 11933	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` B ) = B )
25	20, 24	eqtrd 2271	. . . . . . . . . . . 12 |- ( ph -> ( B gcd B ) = B )
26	25	eqeq1d 2247	. . . . . . . . . . 11 |- ( ph -> ( ( B gcd B ) = 1 <-> B = 1 ) )
27	17, 26	mtbird 684	. . . . . . . . . 10 |- ( ph -> -. ( B gcd B ) = 1 )
28	27	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M e. NN ) -> -. ( B gcd B ) = 1 )
29	18	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ M e. NN ) -> B e. ZZ )
30		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ M e. NN ) -> M e. NN )
31		rpexp 12931	. . . . . . . . . 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 1278	. . . . . . . . 9 |- ( ( ph /\ M e. NN ) -> ( ( ( B ^ M ) gcd B ) = 1 <-> ( B gcd B ) = 1 ) )
33	28, 32	mtbird 684	. . . . . . . 8 |- ( ( ph /\ M e. NN ) -> -. ( ( B ^ M ) gcd B ) = 1 )
34	33	adantr 276	. . . . . . 7 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> -. ( ( B ^ M ) gcd B ) = 1 )
35		oveq1 6092	. . . . . . . . . 10 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( X ^ N ) gcd B ) = ( ( B ^ M ) gcd B ) )
36	35	eqeq1d 2247	. . . . . . . . 9 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
37	36	eqcoms 2241	. . . . . . . 8 |- ( ( B ^ M ) = ( X ^ N ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
38	37	adantl 277	. . . . . . 7 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
39	34, 38	mtbird 684	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> -. ( ( X ^ N ) gcd B ) = 1 )
40	12, 39	pm2.65da 671	. . . . 5 |- ( ( ph /\ M e. NN ) -> -. ( B ^ M ) = ( X ^ N ) )
41	40	neqcomd 2243	. . . 4 |- ( ( ph /\ M e. NN ) -> -. ( X ^ N ) = ( B ^ M ) )
42	41	neqned 2427	. . 3 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) =/= ( B ^ M ) )
43	4	nnzd 9767	. . . . . 6 |- ( ph -> X e. ZZ )
44	43	adantr 276	. . . . 5 |- ( ( ph /\ M e. NN ) -> X e. ZZ )
45	8	nnnn0d 9620	. . . . . 6 |- ( ph -> N e. NN0 )
46	45	adantr 276	. . . . 5 |- ( ( ph /\ M e. NN ) -> N e. NN0 )
47		zexpcl 10991	. . . . 5 |- ( ( X e. ZZ /\ N e. NN0 ) -> ( X ^ N ) e. ZZ )
48	44, 46, 47	syl2anc 415	. . . 4 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) e. ZZ )
49	30	nnnn0d 9620	. . . . 5 |- ( ( ph /\ M e. NN ) -> M e. NN0 )
50		zexpcl 10991	. . . . 5 |- ( ( B e. ZZ /\ M e. NN0 ) -> ( B ^ M ) e. ZZ )
51	29, 49, 50	syl2anc 415	. . . 4 |- ( ( ph /\ M e. NN ) -> ( B ^ M ) e. ZZ )
52		zapne 9719	. . . 4 |- ( ( ( X ^ N ) e. ZZ /\ ( B ^ M ) e. ZZ ) -> ( ( X ^ N ) # ( B ^ M ) <-> ( X ^ N ) =/= ( B ^ M ) ) )
53	48, 51, 52	syl2anc 415	. . 3 |- ( ( ph /\ M e. NN ) -> ( ( X ^ N ) # ( B ^ M ) <-> ( X ^ N ) =/= ( B ^ M ) ) )
54	42, 53	mpbird 167	. 2 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) # ( B ^ M ) )
55	7	nnrpd 10095	. . . . . 6 |- ( ph -> B e. RR+ )
56	55	adantr 276	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> B e. RR+ )
57		logbgcd1irraplem.m	. . . . . 6 |- ( ph -> M e. ZZ )
58	57	adantr 276	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> M e. ZZ )
59	56, 58	rpexpcld 11135	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR+ )
60	59	rpred 10097	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR )
61	4	nnred 9317	. . . . 5 |- ( ph -> X e. RR )
62	61, 45	reexpcld 11128	. . . 4 |- ( ph -> ( X ^ N ) e. RR )
63	62	adantr 276	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) e. RR )
64		1red 8341	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR )
65		1rp 10058	. . . . . . 7 |- 1 e. RR+
66	65	a1i 9	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR+ )
67	21	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. RR )
68		simpr 110	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> -u M e. NN0 )
69	7	nnge1d 9347	. . . . . . . . 9 |- ( ph -> 1 <_ B )
70	69	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ B )
71	67, 68, 70	expge1d 11130	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( B ^ -u M ) )
72	67	recnd 8354	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. CC )
73	7	nnap0d 9350	. . . . . . . . 9 |- ( ph -> B # 0 )
74	73	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B # 0 )
75	72, 74, 58	expnegapd 11118	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ -u M ) = ( 1 / ( B ^ M ) ) )
76	71, 75	breqtrd 4156	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( 1 / ( B ^ M ) ) )
77	66, 59, 76	lerec2d 10119	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ ( 1 / 1 ) )
78		1div1e1 9034	. . . . 5 |- ( 1 / 1 ) = 1
79	77, 78	breqtrdi 4171	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ 1 )
80		eluz2gt1 10002	. . . . . . 7 |- ( X e. ( ZZ>= ` 2 ) -> 1 < X )
81	2, 80	syl 14	. . . . . 6 |- ( ph -> 1 < X )
82		expgt1 11014	. . . . . 6 |- ( ( X e. RR /\ N e. NN /\ 1 < X ) -> 1 < ( X ^ N ) )
83	61, 8, 81, 82	syl3anc 1278	. . . . 5 |- ( ph -> 1 < ( X ^ N ) )
84	83	adantr 276	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 < ( X ^ N ) )
85	60, 64, 63, 79, 84	lelttrd 8451	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) < ( X ^ N ) )
86	60, 63, 85	gtapd 8965	. 2 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) # ( B ^ M ) )
87		elznn 9660	. . . 4 |- ( M e. ZZ <-> ( M e. RR /\ ( M e. NN \/ -u M e. NN0 ) ) )
88	57, 87	sylib 122	. . 3 |- ( ph -> ( M e. RR /\ ( M e. NN \/ -u M e. NN0 ) ) )
89	88	simprd 114	. 2 |- ( ph -> ( M e. NN \/ -u M e. NN0 ) )
90	54, 86, 89	mpjaodan 810	1 |- ( ph -> ( X ^ N ) # ( B ^ M ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   < clt 8360   <_ cle 8361   -ucneg 8498   # cap 8909   / cdiv 9002   NNcn 9304   2c2 9355   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   RR+crp 10054   ^cexp 10975   abscabs 11763   gcd cgcd 12730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886

This theorem is used by:  logbgcd1irraplemap  16071