Theorem logbgcd1irraplemexp 15657

Description: Lemma for logbgcd1irrap 15659. 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 9773	. . . . . . . . . 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 9773	. . . . . . . . . 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 12563	. . . . . . . . 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 1271	. . . . . . . 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 488	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> ( ( X ^ N ) gcd B ) = 1 )
13		1red 8172	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR )
14		eluz2gt1 9809	. . . . . . . . . . . . . 14 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
15	5, 14	syl 14	. . . . . . . . . . . . 13 |- ( ph -> 1 < B )
16	13, 15	gtned 8270	. . . . . . . . . . . 12 |- ( ph -> B =/= 1 )
17	16	neneqd 2421	. . . . . . . . . . 11 |- ( ph -> -. B = 1 )
18	7	nnzd 9579	. . . . . . . . . . . . . 14 |- ( ph -> B e. ZZ )
19		gcdid 12522	. . . . . . . . . . . . . 14 |- ( B e. ZZ -> ( B gcd B ) = ( abs ` B ) )
20	18, 19	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( B gcd B ) = ( abs ` B ) )
21	7	nnred 9134	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
22	7	nnnn0d 9433	. . . . . . . . . . . . . . 15 |- ( ph -> B e. NN0 )
23	22	nn0ge0d 9436	. . . . . . . . . . . . . 14 |- ( ph -> 0 <_ B )
24	21, 23	absidd 11693	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` B ) = B )
25	20, 24	eqtrd 2262	. . . . . . . . . . . 12 |- ( ph -> ( B gcd B ) = B )
26	25	eqeq1d 2238	. . . . . . . . . . 11 |- ( ph -> ( ( B gcd B ) = 1 <-> B = 1 ) )
27	17, 26	mtbird 677	. . . . . . . . . 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 12690	. . . . . . . . . 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 1271	. . . . . . . . 9 |- ( ( ph /\ M e. NN ) -> ( ( ( B ^ M ) gcd B ) = 1 <-> ( B gcd B ) = 1 ) )
33	28, 32	mtbird 677	. . . . . . . 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 6014	. . . . . . . . . 10 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( X ^ N ) gcd B ) = ( ( B ^ M ) gcd B ) )
36	35	eqeq1d 2238	. . . . . . . . 9 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
37	36	eqcoms 2232	. . . . . . . 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 677	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> -. ( ( X ^ N ) gcd B ) = 1 )
40	12, 39	pm2.65da 665	. . . . 5 |- ( ( ph /\ M e. NN ) -> -. ( B ^ M ) = ( X ^ N ) )
41	40	neqcomd 2234	. . . 4 |- ( ( ph /\ M e. NN ) -> -. ( X ^ N ) = ( B ^ M ) )
42	41	neqned 2407	. . 3 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) =/= ( B ^ M ) )
43	4	nnzd 9579	. . . . . 6 |- ( ph -> X e. ZZ )
44	43	adantr 276	. . . . 5 |- ( ( ph /\ M e. NN ) -> X e. ZZ )
45	8	nnnn0d 9433	. . . . . 6 |- ( ph -> N e. NN0 )
46	45	adantr 276	. . . . 5 |- ( ( ph /\ M e. NN ) -> N e. NN0 )
47		zexpcl 10788	. . . . 5 |- ( ( X e. ZZ /\ N e. NN0 ) -> ( X ^ N ) e. ZZ )
48	44, 46, 47	syl2anc 411	. . . 4 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) e. ZZ )
49	30	nnnn0d 9433	. . . . 5 |- ( ( ph /\ M e. NN ) -> M e. NN0 )
50		zexpcl 10788	. . . . 5 |- ( ( B e. ZZ /\ M e. NN0 ) -> ( B ^ M ) e. ZZ )
51	29, 49, 50	syl2anc 411	. . . 4 |- ( ( ph /\ M e. NN ) -> ( B ^ M ) e. ZZ )
52		zapne 9532	. . . 4 |- ( ( ( X ^ N ) e. ZZ /\ ( B ^ M ) e. ZZ ) -> ( ( X ^ N ) # ( B ^ M ) <-> ( X ^ N ) =/= ( B ^ M ) ) )
53	48, 51, 52	syl2anc 411	. . 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 9902	. . . . . 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 10931	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR+ )
60	59	rpred 9904	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR )
61	4	nnred 9134	. . . . 5 |- ( ph -> X e. RR )
62	61, 45	reexpcld 10924	. . . 4 |- ( ph -> ( X ^ N ) e. RR )
63	62	adantr 276	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) e. RR )
64		1red 8172	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR )
65		1rp 9865	. . . . . . 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 9164	. . . . . . . . 9 |- ( ph -> 1 <_ B )
70	69	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ B )
71	67, 68, 70	expge1d 10926	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( B ^ -u M ) )
72	67	recnd 8186	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. CC )
73	7	nnap0d 9167	. . . . . . . . 9 |- ( ph -> B # 0 )
74	73	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B # 0 )
75	72, 74, 58	expnegapd 10914	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ -u M ) = ( 1 / ( B ^ M ) ) )
76	71, 75	breqtrd 4109	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( 1 / ( B ^ M ) ) )
77	66, 59, 76	lerec2d 9926	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ ( 1 / 1 ) )
78		1div1e1 8862	. . . . 5 |- ( 1 / 1 ) = 1
79	77, 78	breqtrdi 4124	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ 1 )
80		eluz2gt1 9809	. . . . . . 7 |- ( X e. ( ZZ>= ` 2 ) -> 1 < X )
81	2, 80	syl 14	. . . . . 6 |- ( ph -> 1 < X )
82		expgt1 10811	. . . . . 6 |- ( ( X e. RR /\ N e. NN /\ 1 < X ) -> 1 < ( X ^ N ) )
83	61, 8, 81, 82	syl3anc 1271	. . . . 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 8282	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) < ( X ^ N ) )
86	60, 63, 85	gtapd 8795	. 2 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) # ( B ^ M ) )
87		elznn 9473	. . . 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 803	1 |- ( ph -> ( X ^ N ) # ( B ^ M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   RRcr 8009   0cc0 8010   1c1 8011   < clt 8192   <_ cle 8193   -ucneg 8329   # cap 8739   / cdiv 8830   NNcn 9121   2c2 9172   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   RR+crp 9861   ^cexp 10772   abscabs 11523   gcd cgcd 12489

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7162  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-dvds 12314  df-gcd 12490  df-prm 12645

This theorem is referenced by:  logbgcd1irraplemap  15658