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Theorem rpabscxpbnd 15630

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 15584	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5633	. . 3
6	1	relogcld 15572	. . . . . 6
7	6	recnd 8186	. . . . 5
8	2, 7	mulcld 8178	. . . 4
9		absef 12297	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11465	. . . . . . 7
12	7	recld 11465	. . . . . . 7
13	11, 12	remulcld 8188	. . . . . 6
14	13	recnd 8186	. . . . 5
15	2	imcld 11466	. . . . . . 7
16	7	imcld 11466	. . . . . . . 8
17	16	renegcld 8537	. . . . . . 7
18	15, 17	remulcld 8188	. . . . . 6
19	18	recnd 8186	. . . . 5
20		efadd 12202	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8188	. . . . . . . 8
23	22	recnd 8186	. . . . . . 7
24	14, 23	negsubd 8474	. . . . . 6
25	15	recnd 8186	. . . . . . . 8
26	16	recnd 8186	. . . . . . . 8
27	25, 26	mulneg2d 8569	. . . . . . 7
28	27	oveq2d 6023	. . . . . 6
29	2, 7	remuld 11490	. . . . . 6
30	24, 28, 29	3eqtr4d 2272	. . . . 5
31	30	fveq2d 5633	. . . 4
32	6	rered 11496	. . . . . . . . 9
33	1	rpred 9904	. . . . . . . . . . 11
34	1	rpge0d 9908	. . . . . . . . . . 11
35	33, 34	absidd 11694	. . . . . . . . . 10
36	35	fveq2d 5633	. . . . . . . . 9
37	32, 36	eqtr4d 2265	. . . . . . . 8
38	37	oveq2d 6023	. . . . . . 7
39	38	fveq2d 5633	. . . . . 6
40	35, 1	eqeltrd 2306	. . . . . . 7
41	11	recnd 8186	. . . . . . 7
42		rpcxpef 15584	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2265	. . . . 5
45	44	oveq1d 6022	. . . 4
46	21, 31, 45	3eqtr3d 2270	. . 3
47	5, 10, 46	3eqtrd 2266	. 2
48	40, 11	rpcxpcld 15623	. . . . 5
49	48	rpred 9904	. . . 4
50	18	reefcld 12196	. . . 4
51	49, 50	remulcld 8188	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9944	. . . . . 6
55	54, 11	rpcxpcld 15623	. . . . 5
56	55	rpred 9904	. . . 4
57	56, 50	remulcld 8188	. . 3
58	2	abscld 11708	. . . . . 6
59		pire 15476	. . . . . 6
60		remulcl 8138	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12196	. . . 4
63	56, 62	remulcld 8188	. . 3
64	18	rpefcld 12213	. . . . 5
65	64	rpge0d 9908	. . . 4
66	1	rpcnd 9906	. . . . . . 7
67	1	rpap0d 9910	. . . . . . 7 #
68	66, 67	absrpclapd 11715	. . . . . 6
69	52, 68, 53	rpgecld 9944	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9901	. . . . . 6
72		rpcxple2 15608	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1271	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9097	. . 3
76	55	rpge0d 9908	. . . 4
77	25	abscld 11708	. . . . . . 7
78	17	recnd 8186	. . . . . . . 8
79	78	abscld 11708	. . . . . . 7
80	77, 79	remulcld 8188	. . . . . 6
81	18	leabsd 11688	. . . . . . 7
82	25, 78	absmuld 11721	. . . . . . 7
83	81, 82	breqtrd 4109	. . . . . 6
84	58, 79	remulcld 8188	. . . . . . 7
85	78	absge0d 11711	. . . . . . . 8
86		absimle 11611	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9097	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11711	. . . . . . . 8
91	26	absnegd 11716	. . . . . . . . 9
92	59	renegcli 8419	. . . . . . . . . . . 12
93		0re 8157	. . . . . . . . . . . 12
94		pipos 15478	. . . . . . . . . . . . 13
95		lt0neg2 8627	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8260	. . . . . . . . . . 11
99	6	reim0d 11497	. . . . . . . . . . 11
100	98, 99	breqtrrid 4121	. . . . . . . . . 10
101	93, 59, 94	ltleii 8260	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4122	. . . . . . . . . 10
103		absle 11616	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 950	. . . . . . . . 9
106	91, 105	eqbrtrd 4105	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9098	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8281	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8281	. . . . 5
110		efle 15466	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9098	. . 3
114	51, 57, 63, 75, 113	letrd 8281	. 2
115	47, 114	eqbrtrd 4105	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1395

wcel 2200 class class class wbr 4083

cfv 5318 (class class class)co 6007

cc 8008

cr 8009

cc0 8010

caddc 8013

cmul 8015

clt 8192

cle 8193

cmin 8328

cneg 8329

crp 9861

cre 11367

cim 11368

cabs 11524

ce 12169

cpi 12174

clog 15546

ccxp 15547

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 ax-pre-suploc 8131 ax-addf 8132 ax-mulf 8133

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-disj 4060 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-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7162 df-inf 7163 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-5 9183 df-6 9184 df-7 9185 df-8 9186 df-9 9187 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-xneg 9980 df-xadd 9981 df-ioo 10100 df-ioc 10101 df-ico 10102 df-icc 10103 df-fz 10217 df-fzo 10351 df-seqfrec 10682 df-exp 10773 df-fac 10960 df-bc 10982 df-ihash 11010 df-shft 11342 df-cj 11369 df-re 11370 df-im 11371 df-rsqrt 11525 df-abs 11526 df-clim 11806 df-sumdc 11881 df-ef 12175 df-e 12176 df-sin 12177 df-cos 12178 df-pi 12180 df-rest 13290 df-topgen 13309 df-psmet 14523 df-xmet 14524 df-met 14525 df-bl 14526 df-mopn 14527 df-top 14688 df-topon 14701 df-bases 14733 df-ntr 14786 df-cn 14878 df-cnp 14879 df-tx 14943 df-cncf 15261 df-limced 15346 df-dvap 15347 df-relog 15548 df-rpcxp 15549

This theorem is referenced by: (None)

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