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Theorem rpabscxpbnd 15614

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 15568	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5631	. . 3
6	1	relogcld 15556	. . . . . 6
7	6	recnd 8175	. . . . 5
8	2, 7	mulcld 8167	. . . 4
9		absef 12281	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11449	. . . . . . 7
12	7	recld 11449	. . . . . . 7
13	11, 12	remulcld 8177	. . . . . 6
14	13	recnd 8175	. . . . 5
15	2	imcld 11450	. . . . . . 7
16	7	imcld 11450	. . . . . . . 8
17	16	renegcld 8526	. . . . . . 7
18	15, 17	remulcld 8177	. . . . . 6
19	18	recnd 8175	. . . . 5
20		efadd 12186	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8177	. . . . . . . 8
23	22	recnd 8175	. . . . . . 7
24	14, 23	negsubd 8463	. . . . . 6
25	15	recnd 8175	. . . . . . . 8
26	16	recnd 8175	. . . . . . . 8
27	25, 26	mulneg2d 8558	. . . . . . 7
28	27	oveq2d 6017	. . . . . 6
29	2, 7	remuld 11474	. . . . . 6
30	24, 28, 29	3eqtr4d 2272	. . . . 5
31	30	fveq2d 5631	. . . 4
32	6	rered 11480	. . . . . . . . 9
33	1	rpred 9892	. . . . . . . . . . 11
34	1	rpge0d 9896	. . . . . . . . . . 11
35	33, 34	absidd 11678	. . . . . . . . . 10
36	35	fveq2d 5631	. . . . . . . . 9
37	32, 36	eqtr4d 2265	. . . . . . . 8
38	37	oveq2d 6017	. . . . . . 7
39	38	fveq2d 5631	. . . . . 6
40	35, 1	eqeltrd 2306	. . . . . . 7
41	11	recnd 8175	. . . . . . 7
42		rpcxpef 15568	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2265	. . . . 5
45	44	oveq1d 6016	. . . 4
46	21, 31, 45	3eqtr3d 2270	. . 3
47	5, 10, 46	3eqtrd 2266	. 2
48	40, 11	rpcxpcld 15607	. . . . 5
49	48	rpred 9892	. . . 4
50	18	reefcld 12180	. . . 4
51	49, 50	remulcld 8177	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9932	. . . . . 6
55	54, 11	rpcxpcld 15607	. . . . 5
56	55	rpred 9892	. . . 4
57	56, 50	remulcld 8177	. . 3
58	2	abscld 11692	. . . . . 6
59		pire 15460	. . . . . 6
60		remulcl 8127	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12180	. . . 4
63	56, 62	remulcld 8177	. . 3
64	18	rpefcld 12197	. . . . 5
65	64	rpge0d 9896	. . . 4
66	1	rpcnd 9894	. . . . . . 7
67	1	rpap0d 9898	. . . . . . 7 #
68	66, 67	absrpclapd 11699	. . . . . 6
69	52, 68, 53	rpgecld 9932	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9889	. . . . . 6
72		rpcxple2 15592	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1271	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9086	. . 3
76	55	rpge0d 9896	. . . 4
77	25	abscld 11692	. . . . . . 7
78	17	recnd 8175	. . . . . . . 8
79	78	abscld 11692	. . . . . . 7
80	77, 79	remulcld 8177	. . . . . 6
81	18	leabsd 11672	. . . . . . 7
82	25, 78	absmuld 11705	. . . . . . 7
83	81, 82	breqtrd 4109	. . . . . 6
84	58, 79	remulcld 8177	. . . . . . 7
85	78	absge0d 11695	. . . . . . . 8
86		absimle 11595	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9086	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11695	. . . . . . . 8
91	26	absnegd 11700	. . . . . . . . 9
92	59	renegcli 8408	. . . . . . . . . . . 12
93		0re 8146	. . . . . . . . . . . 12
94		pipos 15462	. . . . . . . . . . . . 13
95		lt0neg2 8616	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8249	. . . . . . . . . . 11
99	6	reim0d 11481	. . . . . . . . . . 11
100	98, 99	breqtrrid 4121	. . . . . . . . . 10
101	93, 59, 94	ltleii 8249	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4122	. . . . . . . . . 10
103		absle 11600	. . . . . . . . . . 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 9087	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8270	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8270	. . . . 5
110		efle 15450	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9087	. . 3
114	51, 57, 63, 75, 113	letrd 8270	. 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 6001

cc 7997

cr 7998

cc0 7999

caddc 8002

cmul 8004

clt 8181

cle 8182

cmin 8317

cneg 8318

crp 9849

cre 11351

cim 11352

cabs 11508

ce 12153

cpi 12158

clog 15530

ccxp 15531

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 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119 ax-pre-suploc 8120 ax-addf 8121 ax-mulf 8122

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 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-of 6218 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-frec 6537 df-1o 6562 df-oadd 6566 df-er 6680 df-map 6797 df-pm 6798 df-en 6888 df-dom 6889 df-fin 6890 df-sup 7151 df-inf 7152 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-5 9172 df-6 9173 df-7 9174 df-8 9175 df-9 9176 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-xneg 9968 df-xadd 9969 df-ioo 10088 df-ioc 10089 df-ico 10090 df-icc 10091 df-fz 10205 df-fzo 10339 df-seqfrec 10670 df-exp 10761 df-fac 10948 df-bc 10970 df-ihash 10998 df-shft 11326 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-clim 11790 df-sumdc 11865 df-ef 12159 df-e 12160 df-sin 12161 df-cos 12162 df-pi 12164 df-rest 13274 df-topgen 13293 df-psmet 14507 df-xmet 14508 df-met 14509 df-bl 14510 df-mopn 14511 df-top 14672 df-topon 14685 df-bases 14717 df-ntr 14770 df-cn 14862 df-cnp 14863 df-tx 14927 df-cncf 15245 df-limced 15330 df-dvap 15331 df-relog 15532 df-rpcxp 15533

This theorem is referenced by: (None)

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