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Theorem rpabscxpbnd 15527

Description: Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)

**Hypotheses**
Ref	Expression
rpabscxpbnd.1
abscxpbnd.2
rpabscxpbnd.3
abscxpbnd.4
abscxpbnd.5

**Assertion**
Ref	Expression
rpabscxpbnd

**Proof of Theorem rpabscxpbnd**
Step	Hyp	Ref	Expression
1		rpabscxpbnd.1	. . . . 5
2		abscxpbnd.2	. . . . 5
3		rpcxpef 15481	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5603	. . 3
6	1	relogcld 15469	. . . . . 6
7	6	recnd 8136	. . . . 5
8	2, 7	mulcld 8128	. . . 4
9		absef 12196	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11364	. . . . . . 7
12	7	recld 11364	. . . . . . 7
13	11, 12	remulcld 8138	. . . . . 6
14	13	recnd 8136	. . . . 5
15	2	imcld 11365	. . . . . . 7
16	7	imcld 11365	. . . . . . . 8
17	16	renegcld 8487	. . . . . . 7
18	15, 17	remulcld 8138	. . . . . 6
19	18	recnd 8136	. . . . 5
20		efadd 12101	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8138	. . . . . . . 8
23	22	recnd 8136	. . . . . . 7
24	14, 23	negsubd 8424	. . . . . 6
25	15	recnd 8136	. . . . . . . 8
26	16	recnd 8136	. . . . . . . 8
27	25, 26	mulneg2d 8519	. . . . . . 7
28	27	oveq2d 5983	. . . . . 6
29	2, 7	remuld 11389	. . . . . 6
30	24, 28, 29	3eqtr4d 2250	. . . . 5
31	30	fveq2d 5603	. . . 4
32	6	rered 11395	. . . . . . . . 9
33	1	rpred 9853	. . . . . . . . . . 11
34	1	rpge0d 9857	. . . . . . . . . . 11
35	33, 34	absidd 11593	. . . . . . . . . 10
36	35	fveq2d 5603	. . . . . . . . 9
37	32, 36	eqtr4d 2243	. . . . . . . 8
38	37	oveq2d 5983	. . . . . . 7
39	38	fveq2d 5603	. . . . . 6
40	35, 1	eqeltrd 2284	. . . . . . 7
41	11	recnd 8136	. . . . . . 7
42		rpcxpef 15481	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2243	. . . . 5
45	44	oveq1d 5982	. . . 4
46	21, 31, 45	3eqtr3d 2248	. . 3
47	5, 10, 46	3eqtrd 2244	. 2
48	40, 11	rpcxpcld 15520	. . . . 5
49	48	rpred 9853	. . . 4
50	18	reefcld 12095	. . . 4
51	49, 50	remulcld 8138	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9893	. . . . . 6
55	54, 11	rpcxpcld 15520	. . . . 5
56	55	rpred 9853	. . . 4
57	56, 50	remulcld 8138	. . 3
58	2	abscld 11607	. . . . . 6
59		pire 15373	. . . . . 6
60		remulcl 8088	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12095	. . . 4
63	56, 62	remulcld 8138	. . 3
64	18	rpefcld 12112	. . . . 5
65	64	rpge0d 9857	. . . 4
66	1	rpcnd 9855	. . . . . . 7
67	1	rpap0d 9859	. . . . . . 7 #
68	66, 67	absrpclapd 11614	. . . . . 6
69	52, 68, 53	rpgecld 9893	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9850	. . . . . 6
72		rpcxple2 15505	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1250	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9047	. . 3
76	55	rpge0d 9857	. . . 4
77	25	abscld 11607	. . . . . . 7
78	17	recnd 8136	. . . . . . . 8
79	78	abscld 11607	. . . . . . 7
80	77, 79	remulcld 8138	. . . . . 6
81	18	leabsd 11587	. . . . . . 7
82	25, 78	absmuld 11620	. . . . . . 7
83	81, 82	breqtrd 4085	. . . . . 6
84	58, 79	remulcld 8138	. . . . . . 7
85	78	absge0d 11610	. . . . . . . 8
86		absimle 11510	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9047	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11610	. . . . . . . 8
91	26	absnegd 11615	. . . . . . . . 9
92	59	renegcli 8369	. . . . . . . . . . . 12
93		0re 8107	. . . . . . . . . . . 12
94		pipos 15375	. . . . . . . . . . . . 13
95		lt0neg2 8577	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8210	. . . . . . . . . . 11
99	6	reim0d 11396	. . . . . . . . . . 11
100	98, 99	breqtrrid 4097	. . . . . . . . . 10
101	93, 59, 94	ltleii 8210	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4098	. . . . . . . . . 10
103		absle 11515	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 947	. . . . . . . . 9
106	91, 105	eqbrtrd 4081	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9048	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8231	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8231	. . . . 5
110		efle 15363	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9048	. . 3
114	51, 57, 63, 75, 113	letrd 8231	. 2
115	47, 114	eqbrtrd 4081	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1373

wcel 2178 class class class wbr 4059

cfv 5290 (class class class)co 5967

cc 7958

cr 7959

cc0 7960

caddc 7963

cmul 7965

clt 8142

cle 8143

cmin 8278

cneg 8279

crp 9810

cre 11266

cim 11267

cabs 11423

ce 12068

cpi 12073

clog 15443

ccxp 15444

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080 ax-pre-suploc 8081 ax-addf 8082 ax-mulf 8083

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-disj 4036 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-of 6181 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-frec 6500 df-1o 6525 df-oadd 6529 df-er 6643 df-map 6760 df-pm 6761 df-en 6851 df-dom 6852 df-fin 6853 df-sup 7112 df-inf 7113 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-5 9133 df-6 9134 df-7 9135 df-8 9136 df-9 9137 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-xneg 9929 df-xadd 9930 df-ioo 10049 df-ioc 10050 df-ico 10051 df-icc 10052 df-fz 10166 df-fzo 10300 df-seqfrec 10630 df-exp 10721 df-fac 10908 df-bc 10930 df-ihash 10958 df-shft 11241 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-clim 11705 df-sumdc 11780 df-ef 12074 df-e 12075 df-sin 12076 df-cos 12077 df-pi 12079 df-rest 13188 df-topgen 13207 df-psmet 14420 df-xmet 14421 df-met 14422 df-bl 14423 df-mopn 14424 df-top 14585 df-topon 14598 df-bases 14630 df-ntr 14683 df-cn 14775 df-cnp 14776 df-tx 14840 df-cncf 15158 df-limced 15243 df-dvap 15244 df-relog 15445 df-rpcxp 15446

This theorem is referenced by: (None)

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