Theorem logbgcd1irraplemexp 13433

Description: Lemma for logbgcd1irrap 13435. 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 9495	. . . . . . . . . 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 9495	. . . . . . . . . 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 11945	. . . . . . . . 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 1227	. . . . . . . 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 7905	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR )
14		eluz2gt1 9531	. . . . . . . . . . . . . 14 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
15	5, 14	syl 14	. . . . . . . . . . . . 13 |- ( ph -> 1 < B )
16	13, 15	gtned 8002	. . . . . . . . . . . 12 |- ( ph -> B =/= 1 )
17	16	neneqd 2355	. . . . . . . . . . 11 |- ( ph -> -. B = 1 )
18	7	nnzd 9303	. . . . . . . . . . . . . 14 |- ( ph -> B e. ZZ )
19		gcdid 11904	. . . . . . . . . . . . . 14 |- ( B e. ZZ -> ( B gcd B ) = ( abs ` B ) )
20	18, 19	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( B gcd B ) = ( abs ` B ) )
21	7	nnred 8861	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
22	7	nnnn0d 9158	. . . . . . . . . . . . . . 15 |- ( ph -> B e. NN0 )
23	22	nn0ge0d 9161	. . . . . . . . . . . . . 14 |- ( ph -> 0 <_ B )
24	21, 23	absidd 11095	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` B ) = B )
25	20, 24	eqtrd 2197	. . . . . . . . . . . 12 |- ( ph -> ( B gcd B ) = B )
26	25	eqeq1d 2173	. . . . . . . . . . 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 12064	. . . . . . . . . 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 1227	. . . . . . . . 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 5843	. . . . . . . . . 10 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( X ^ N ) gcd B ) = ( ( B ^ M ) gcd B ) )
36	35	eqeq1d 2173	. . . . . . . . 9 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
37	36	eqcoms 2167	. . . . . . . 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 2169	. . . 4 |- ( ( ph /\ M e. NN ) -> -. ( X ^ N ) = ( B ^ M ) )
42	41	neqned 2341	. . 3 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) =/= ( B ^ M ) )
43	4	nnzd 9303	. . . . . 6 |- ( ph -> X e. ZZ )
44	43	adantr 274	. . . . 5 |- ( ( ph /\ M e. NN ) -> X e. ZZ )
45	8	nnnn0d 9158	. . . . . 6 |- ( ph -> N e. NN0 )
46	45	adantr 274	. . . . 5 |- ( ( ph /\ M e. NN ) -> N e. NN0 )
47		zexpcl 10460	. . . . 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 9158	. . . . 5 |- ( ( ph /\ M e. NN ) -> M e. NN0 )
50		zexpcl 10460	. . . . 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 9256	. . . 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 9621	. . . . . 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 10601	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR+ )
60	59	rpred 9623	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR )
61	4	nnred 8861	. . . . 5 |- ( ph -> X e. RR )
62	61, 45	reexpcld 10594	. . . 4 |- ( ph -> ( X ^ N ) e. RR )
63	62	adantr 274	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) e. RR )
64		1red 7905	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR )
65		1rp 9584	. . . . . . 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 8891	. . . . . . . . 9 |- ( ph -> 1 <_ B )
70	69	adantr 274	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ B )
71	67, 68, 70	expge1d 10596	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( B ^ -u M ) )
72	67	recnd 7918	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. CC )
73	7	nnap0d 8894	. . . . . . . . 9 |- ( ph -> B # 0 )
74	73	adantr 274	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B # 0 )
75	72, 74, 58	expnegapd 10584	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ -u M ) = ( 1 / ( B ^ M ) ) )
76	71, 75	breqtrd 4002	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( 1 / ( B ^ M ) ) )
77	66, 59, 76	lerec2d 9645	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ ( 1 / 1 ) )
78		1div1e1 8591	. . . . 5 |- ( 1 / 1 ) = 1
79	77, 78	breqtrdi 4017	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ 1 )
80		eluz2gt1 9531	. . . . . . 7 |- ( X e. ( ZZ>= ` 2 ) -> 1 < X )
81	2, 80	syl 14	. . . . . 6 |- ( ph -> 1 < X )
82		expgt1 10483	. . . . . 6 |- ( ( X e. RR /\ N e. NN /\ 1 < X ) -> 1 < ( X ^ N ) )
83	61, 8, 81, 82	syl3anc 1227	. . . . 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 8014	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) < ( X ^ N ) )
86	60, 63, 85	gtapd 8526	. 2 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) # ( B ^ M ) )
87		elznn 9198	. . . 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 1342   e. wcel 2135   =/= wne 2334   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   RRcr 7743   0cc0 7744   1c1 7745   < clt 7924   <_ cle 7925   -ucneg 8061   # cap 8470   / cdiv 8559   NNcn 8848   2c2 8899   NN0cn0 9105   ZZcz 9182   ZZ>=cuz 9457   RR+crp 9580   ^cexp 10444   abscabs 10925   gcd cgcd 11860

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-1o 6375  df-2o 6376  df-er 6492  df-en 6698  df-sup 6940  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-fz 9936  df-fzo 10068  df-fl 10195  df-mod 10248  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-dvds 11714  df-gcd 11861  df-prm 12019

This theorem is referenced by:  logbgcd1irraplemap  13434