Theorem logbgcd1irraplemexp 15141

Description: Lemma for logbgcd1irrap 15143. 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 9634	. . . . . . . . . 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 9634	. . . . . . . . . 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 12167	. . . . . . . . 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 1249	. . . . . . . 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 8036	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR )
14		eluz2gt1 9670	. . . . . . . . . . . . . 14 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
15	5, 14	syl 14	. . . . . . . . . . . . 13 |- ( ph -> 1 < B )
16	13, 15	gtned 8134	. . . . . . . . . . . 12 |- ( ph -> B =/= 1 )
17	16	neneqd 2385	. . . . . . . . . . 11 |- ( ph -> -. B = 1 )
18	7	nnzd 9441	. . . . . . . . . . . . . 14 |- ( ph -> B e. ZZ )
19		gcdid 12126	. . . . . . . . . . . . . 14 |- ( B e. ZZ -> ( B gcd B ) = ( abs ` B ) )
20	18, 19	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( B gcd B ) = ( abs ` B ) )
21	7	nnred 8997	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
22	7	nnnn0d 9296	. . . . . . . . . . . . . . 15 |- ( ph -> B e. NN0 )
23	22	nn0ge0d 9299	. . . . . . . . . . . . . 14 |- ( ph -> 0 <_ B )
24	21, 23	absidd 11314	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` B ) = B )
25	20, 24	eqtrd 2226	. . . . . . . . . . . 12 |- ( ph -> ( B gcd B ) = B )
26	25	eqeq1d 2202	. . . . . . . . . . 11 |- ( ph -> ( ( B gcd B ) = 1 <-> B = 1 ) )
27	17, 26	mtbird 674	. . . . . . . . . 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 12294	. . . . . . . . . 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 1249	. . . . . . . . 9 |- ( ( ph /\ M e. NN ) -> ( ( ( B ^ M ) gcd B ) = 1 <-> ( B gcd B ) = 1 ) )
33	28, 32	mtbird 674	. . . . . . . 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 5926	. . . . . . . . . 10 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( X ^ N ) gcd B ) = ( ( B ^ M ) gcd B ) )
36	35	eqeq1d 2202	. . . . . . . . 9 |- ( ( X ^ N ) = ( B ^ M ) -> ( ( ( X ^ N ) gcd B ) = 1 <-> ( ( B ^ M ) gcd B ) = 1 ) )
37	36	eqcoms 2196	. . . . . . . 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 674	. . . . . 6 |- ( ( ( ph /\ M e. NN ) /\ ( B ^ M ) = ( X ^ N ) ) -> -. ( ( X ^ N ) gcd B ) = 1 )
40	12, 39	pm2.65da 662	. . . . 5 |- ( ( ph /\ M e. NN ) -> -. ( B ^ M ) = ( X ^ N ) )
41	40	neqcomd 2198	. . . 4 |- ( ( ph /\ M e. NN ) -> -. ( X ^ N ) = ( B ^ M ) )
42	41	neqned 2371	. . 3 |- ( ( ph /\ M e. NN ) -> ( X ^ N ) =/= ( B ^ M ) )
43	4	nnzd 9441	. . . . . 6 |- ( ph -> X e. ZZ )
44	43	adantr 276	. . . . 5 |- ( ( ph /\ M e. NN ) -> X e. ZZ )
45	8	nnnn0d 9296	. . . . . 6 |- ( ph -> N e. NN0 )
46	45	adantr 276	. . . . 5 |- ( ( ph /\ M e. NN ) -> N e. NN0 )
47		zexpcl 10628	. . . . 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 9296	. . . . 5 |- ( ( ph /\ M e. NN ) -> M e. NN0 )
50		zexpcl 10628	. . . . 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 9394	. . . 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 9763	. . . . . 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 10771	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR+ )
60	59	rpred 9765	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) e. RR )
61	4	nnred 8997	. . . . 5 |- ( ph -> X e. RR )
62	61, 45	reexpcld 10764	. . . 4 |- ( ph -> ( X ^ N ) e. RR )
63	62	adantr 276	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) e. RR )
64		1red 8036	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> 1 e. RR )
65		1rp 9726	. . . . . . 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 9027	. . . . . . . . 9 |- ( ph -> 1 <_ B )
70	69	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ B )
71	67, 68, 70	expge1d 10766	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( B ^ -u M ) )
72	67	recnd 8050	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B e. CC )
73	7	nnap0d 9030	. . . . . . . . 9 |- ( ph -> B # 0 )
74	73	adantr 276	. . . . . . . 8 |- ( ( ph /\ -u M e. NN0 ) -> B # 0 )
75	72, 74, 58	expnegapd 10754	. . . . . . 7 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ -u M ) = ( 1 / ( B ^ M ) ) )
76	71, 75	breqtrd 4056	. . . . . 6 |- ( ( ph /\ -u M e. NN0 ) -> 1 <_ ( 1 / ( B ^ M ) ) )
77	66, 59, 76	lerec2d 9787	. . . . 5 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ ( 1 / 1 ) )
78		1div1e1 8725	. . . . 5 |- ( 1 / 1 ) = 1
79	77, 78	breqtrdi 4071	. . . 4 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) <_ 1 )
80		eluz2gt1 9670	. . . . . . 7 |- ( X e. ( ZZ>= ` 2 ) -> 1 < X )
81	2, 80	syl 14	. . . . . 6 |- ( ph -> 1 < X )
82		expgt1 10651	. . . . . 6 |- ( ( X e. RR /\ N e. NN /\ 1 < X ) -> 1 < ( X ^ N ) )
83	61, 8, 81, 82	syl3anc 1249	. . . . 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 8146	. . 3 |- ( ( ph /\ -u M e. NN0 ) -> ( B ^ M ) < ( X ^ N ) )
86	60, 63, 85	gtapd 8658	. 2 |- ( ( ph /\ -u M e. NN0 ) -> ( X ^ N ) # ( B ^ M ) )
87		elznn 9336	. . . 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 799	1 |- ( ph -> ( X ^ N ) # ( B ^ M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   RRcr 7873   0cc0 7874   1c1 7875   < clt 8056   <_ cle 8057   -ucneg 8193   # cap 8602   / cdiv 8693   NNcn 8984   2c2 9035   NN0cn0 9243   ZZcz 9320   ZZ>=cuz 9595   RR+crp 9722   ^cexp 10612   abscabs 11144   gcd cgcd 12082

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-2o 6472  df-er 6589  df-en 6797  df-sup 7045  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083  df-prm 12249

This theorem is referenced by:  logbgcd1irraplemap  15142