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Theorem rpabscxpbnd 15683

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 15637	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5643	. . 3
6	1	relogcld 15625	. . . . . 6
7	6	recnd 8208	. . . . 5
8	2, 7	mulcld 8200	. . . 4
9		absef 12349	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11516	. . . . . . 7
12	7	recld 11516	. . . . . . 7
13	11, 12	remulcld 8210	. . . . . 6
14	13	recnd 8208	. . . . 5
15	2	imcld 11517	. . . . . . 7
16	7	imcld 11517	. . . . . . . 8
17	16	renegcld 8559	. . . . . . 7
18	15, 17	remulcld 8210	. . . . . 6
19	18	recnd 8208	. . . . 5
20		efadd 12254	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8210	. . . . . . . 8
23	22	recnd 8208	. . . . . . 7
24	14, 23	negsubd 8496	. . . . . 6
25	15	recnd 8208	. . . . . . . 8
26	16	recnd 8208	. . . . . . . 8
27	25, 26	mulneg2d 8591	. . . . . . 7
28	27	oveq2d 6034	. . . . . 6
29	2, 7	remuld 11541	. . . . . 6
30	24, 28, 29	3eqtr4d 2274	. . . . 5
31	30	fveq2d 5643	. . . 4
32	6	rered 11547	. . . . . . . . 9
33	1	rpred 9931	. . . . . . . . . . 11
34	1	rpge0d 9935	. . . . . . . . . . 11
35	33, 34	absidd 11745	. . . . . . . . . 10
36	35	fveq2d 5643	. . . . . . . . 9
37	32, 36	eqtr4d 2267	. . . . . . . 8
38	37	oveq2d 6034	. . . . . . 7
39	38	fveq2d 5643	. . . . . 6
40	35, 1	eqeltrd 2308	. . . . . . 7
41	11	recnd 8208	. . . . . . 7
42		rpcxpef 15637	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2267	. . . . 5
45	44	oveq1d 6033	. . . 4
46	21, 31, 45	3eqtr3d 2272	. . 3
47	5, 10, 46	3eqtrd 2268	. 2
48	40, 11	rpcxpcld 15676	. . . . 5
49	48	rpred 9931	. . . 4
50	18	reefcld 12248	. . . 4
51	49, 50	remulcld 8210	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9971	. . . . . 6
55	54, 11	rpcxpcld 15676	. . . . 5
56	55	rpred 9931	. . . 4
57	56, 50	remulcld 8210	. . 3
58	2	abscld 11759	. . . . . 6
59		pire 15529	. . . . . 6
60		remulcl 8160	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12248	. . . 4
63	56, 62	remulcld 8210	. . 3
64	18	rpefcld 12265	. . . . 5
65	64	rpge0d 9935	. . . 4
66	1	rpcnd 9933	. . . . . . 7
67	1	rpap0d 9937	. . . . . . 7 #
68	66, 67	absrpclapd 11766	. . . . . 6
69	52, 68, 53	rpgecld 9971	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9928	. . . . . 6
72		rpcxple2 15661	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1273	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9119	. . 3
76	55	rpge0d 9935	. . . 4
77	25	abscld 11759	. . . . . . 7
78	17	recnd 8208	. . . . . . . 8
79	78	abscld 11759	. . . . . . 7
80	77, 79	remulcld 8210	. . . . . 6
81	18	leabsd 11739	. . . . . . 7
82	25, 78	absmuld 11772	. . . . . . 7
83	81, 82	breqtrd 4114	. . . . . 6
84	58, 79	remulcld 8210	. . . . . . 7
85	78	absge0d 11762	. . . . . . . 8
86		absimle 11662	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9119	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11762	. . . . . . . 8
91	26	absnegd 11767	. . . . . . . . 9
92	59	renegcli 8441	. . . . . . . . . . . 12
93		0re 8179	. . . . . . . . . . . 12
94		pipos 15531	. . . . . . . . . . . . 13
95		lt0neg2 8649	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8282	. . . . . . . . . . 11
99	6	reim0d 11548	. . . . . . . . . . 11
100	98, 99	breqtrrid 4126	. . . . . . . . . 10
101	93, 59, 94	ltleii 8282	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4127	. . . . . . . . . 10
103		absle 11667	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 952	. . . . . . . . 9
106	91, 105	eqbrtrd 4110	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9120	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8303	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8303	. . . . 5
110		efle 15519	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9120	. . 3
114	51, 57, 63, 75, 113	letrd 8303	. 2
115	47, 114	eqbrtrd 4110	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1397

wcel 2202 class class class wbr 4088

cfv 5326 (class class class)co 6018

cc 8030

cr 8031

cc0 8032

caddc 8035

cmul 8037

clt 8214

cle 8215

cmin 8350

cneg 8351

crp 9888

cre 11418

cim 11419

cabs 11575

ce 12221

cpi 12226

clog 15599

ccxp 15600

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-5 9205 df-6 9206 df-7 9207 df-8 9208 df-9 9209 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ioc 10128 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11393 df-cj 11420 df-re 11421 df-im 11422 df-rsqrt 11576 df-abs 11577 df-clim 11857 df-sumdc 11932 df-ef 12227 df-e 12228 df-sin 12229 df-cos 12230 df-pi 12232 df-rest 13342 df-topgen 13361 df-psmet 14576 df-xmet 14577 df-met 14578 df-bl 14579 df-mopn 14580 df-top 14741 df-topon 14754 df-bases 14786 df-ntr 14839 df-cn 14931 df-cnp 14932 df-tx 14996 df-cncf 15314 df-limced 15399 df-dvap 15400 df-relog 15601 df-rpcxp 15602

This theorem is referenced by: (None)

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