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Theorem rpabscxpbnd 15663

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 15617	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5643	. . 3
6	1	relogcld 15605	. . . . . 6
7	6	recnd 8207	. . . . 5
8	2, 7	mulcld 8199	. . . 4
9		absef 12330	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11498	. . . . . . 7
12	7	recld 11498	. . . . . . 7
13	11, 12	remulcld 8209	. . . . . 6
14	13	recnd 8207	. . . . 5
15	2	imcld 11499	. . . . . . 7
16	7	imcld 11499	. . . . . . . 8
17	16	renegcld 8558	. . . . . . 7
18	15, 17	remulcld 8209	. . . . . 6
19	18	recnd 8207	. . . . 5
20		efadd 12235	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8209	. . . . . . . 8
23	22	recnd 8207	. . . . . . 7
24	14, 23	negsubd 8495	. . . . . 6
25	15	recnd 8207	. . . . . . . 8
26	16	recnd 8207	. . . . . . . 8
27	25, 26	mulneg2d 8590	. . . . . . 7
28	27	oveq2d 6033	. . . . . 6
29	2, 7	remuld 11523	. . . . . 6
30	24, 28, 29	3eqtr4d 2274	. . . . 5
31	30	fveq2d 5643	. . . 4
32	6	rered 11529	. . . . . . . . 9
33	1	rpred 9930	. . . . . . . . . . 11
34	1	rpge0d 9934	. . . . . . . . . . 11
35	33, 34	absidd 11727	. . . . . . . . . 10
36	35	fveq2d 5643	. . . . . . . . 9
37	32, 36	eqtr4d 2267	. . . . . . . 8
38	37	oveq2d 6033	. . . . . . 7
39	38	fveq2d 5643	. . . . . 6
40	35, 1	eqeltrd 2308	. . . . . . 7
41	11	recnd 8207	. . . . . . 7
42		rpcxpef 15617	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2267	. . . . 5
45	44	oveq1d 6032	. . . 4
46	21, 31, 45	3eqtr3d 2272	. . 3
47	5, 10, 46	3eqtrd 2268	. 2
48	40, 11	rpcxpcld 15656	. . . . 5
49	48	rpred 9930	. . . 4
50	18	reefcld 12229	. . . 4
51	49, 50	remulcld 8209	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9970	. . . . . 6
55	54, 11	rpcxpcld 15656	. . . . 5
56	55	rpred 9930	. . . 4
57	56, 50	remulcld 8209	. . 3
58	2	abscld 11741	. . . . . 6
59		pire 15509	. . . . . 6
60		remulcl 8159	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12229	. . . 4
63	56, 62	remulcld 8209	. . 3
64	18	rpefcld 12246	. . . . 5
65	64	rpge0d 9934	. . . 4
66	1	rpcnd 9932	. . . . . . 7
67	1	rpap0d 9936	. . . . . . 7 #
68	66, 67	absrpclapd 11748	. . . . . 6
69	52, 68, 53	rpgecld 9970	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9927	. . . . . 6
72		rpcxple2 15641	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1273	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9118	. . 3
76	55	rpge0d 9934	. . . 4
77	25	abscld 11741	. . . . . . 7
78	17	recnd 8207	. . . . . . . 8
79	78	abscld 11741	. . . . . . 7
80	77, 79	remulcld 8209	. . . . . 6
81	18	leabsd 11721	. . . . . . 7
82	25, 78	absmuld 11754	. . . . . . 7
83	81, 82	breqtrd 4114	. . . . . 6
84	58, 79	remulcld 8209	. . . . . . 7
85	78	absge0d 11744	. . . . . . . 8
86		absimle 11644	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9118	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11744	. . . . . . . 8
91	26	absnegd 11749	. . . . . . . . 9
92	59	renegcli 8440	. . . . . . . . . . . 12
93		0re 8178	. . . . . . . . . . . 12
94		pipos 15511	. . . . . . . . . . . . 13
95		lt0neg2 8648	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8281	. . . . . . . . . . 11
99	6	reim0d 11530	. . . . . . . . . . 11
100	98, 99	breqtrrid 4126	. . . . . . . . . 10
101	93, 59, 94	ltleii 8281	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4127	. . . . . . . . . 10
103		absle 11649	. . . . . . . . . . 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 9119	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8302	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8302	. . . . 5
110		efle 15499	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9119	. . 3
114	51, 57, 63, 75, 113	letrd 8302	. 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 6017

cc 8029

cr 8030

cc0 8031

caddc 8034

cmul 8036

clt 8213

cle 8214

cmin 8349

cneg 8350

crp 9887

cre 11400

cim 11401

cabs 11557

ce 12202

cpi 12207

clog 15579

ccxp 15580

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 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151 ax-pre-suploc 8152 ax-addf 8153 ax-mulf 8154

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 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-of 6234 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-frec 6556 df-1o 6581 df-oadd 6585 df-er 6701 df-map 6818 df-pm 6819 df-en 6909 df-dom 6910 df-fin 6911 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205 df-7 9206 df-8 9207 df-9 9208 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-ioo 10126 df-ioc 10127 df-ico 10128 df-icc 10129 df-fz 10243 df-fzo 10377 df-seqfrec 10709 df-exp 10800 df-fac 10987 df-bc 11009 df-ihash 11037 df-shft 11375 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-clim 11839 df-sumdc 11914 df-ef 12208 df-e 12209 df-sin 12210 df-cos 12211 df-pi 12213 df-rest 13323 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-met 14558 df-bl 14559 df-mopn 14560 df-top 14721 df-topon 14734 df-bases 14766 df-ntr 14819 df-cn 14911 df-cnp 14912 df-tx 14976 df-cncf 15294 df-limced 15379 df-dvap 15380 df-relog 15581 df-rpcxp 15582

This theorem is referenced by: (None)

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