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Theorem rpabscxpbnd 15412

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 15366	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5580	. . 3
6	1	relogcld 15354	. . . . . 6
7	6	recnd 8101	. . . . 5
8	2, 7	mulcld 8093	. . . 4
9		absef 12081	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11249	. . . . . . 7
12	7	recld 11249	. . . . . . 7
13	11, 12	remulcld 8103	. . . . . 6
14	13	recnd 8101	. . . . 5
15	2	imcld 11250	. . . . . . 7
16	7	imcld 11250	. . . . . . . 8
17	16	renegcld 8452	. . . . . . 7
18	15, 17	remulcld 8103	. . . . . 6
19	18	recnd 8101	. . . . 5
20		efadd 11986	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8103	. . . . . . . 8
23	22	recnd 8101	. . . . . . 7
24	14, 23	negsubd 8389	. . . . . 6
25	15	recnd 8101	. . . . . . . 8
26	16	recnd 8101	. . . . . . . 8
27	25, 26	mulneg2d 8484	. . . . . . 7
28	27	oveq2d 5960	. . . . . 6
29	2, 7	remuld 11274	. . . . . 6
30	24, 28, 29	3eqtr4d 2248	. . . . 5
31	30	fveq2d 5580	. . . 4
32	6	rered 11280	. . . . . . . . 9
33	1	rpred 9818	. . . . . . . . . . 11
34	1	rpge0d 9822	. . . . . . . . . . 11
35	33, 34	absidd 11478	. . . . . . . . . 10
36	35	fveq2d 5580	. . . . . . . . 9
37	32, 36	eqtr4d 2241	. . . . . . . 8
38	37	oveq2d 5960	. . . . . . 7
39	38	fveq2d 5580	. . . . . 6
40	35, 1	eqeltrd 2282	. . . . . . 7
41	11	recnd 8101	. . . . . . 7
42		rpcxpef 15366	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2241	. . . . 5
45	44	oveq1d 5959	. . . 4
46	21, 31, 45	3eqtr3d 2246	. . 3
47	5, 10, 46	3eqtrd 2242	. 2
48	40, 11	rpcxpcld 15405	. . . . 5
49	48	rpred 9818	. . . 4
50	18	reefcld 11980	. . . 4
51	49, 50	remulcld 8103	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9858	. . . . . 6
55	54, 11	rpcxpcld 15405	. . . . 5
56	55	rpred 9818	. . . 4
57	56, 50	remulcld 8103	. . 3
58	2	abscld 11492	. . . . . 6
59		pire 15258	. . . . . 6
60		remulcl 8053	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 11980	. . . 4
63	56, 62	remulcld 8103	. . 3
64	18	rpefcld 11997	. . . . 5
65	64	rpge0d 9822	. . . 4
66	1	rpcnd 9820	. . . . . . 7
67	1	rpap0d 9824	. . . . . . 7 #
68	66, 67	absrpclapd 11499	. . . . . 6
69	52, 68, 53	rpgecld 9858	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9815	. . . . . 6
72		rpcxple2 15390	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1250	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9012	. . 3
76	55	rpge0d 9822	. . . 4
77	25	abscld 11492	. . . . . . 7
78	17	recnd 8101	. . . . . . . 8
79	78	abscld 11492	. . . . . . 7
80	77, 79	remulcld 8103	. . . . . 6
81	18	leabsd 11472	. . . . . . 7
82	25, 78	absmuld 11505	. . . . . . 7
83	81, 82	breqtrd 4070	. . . . . 6
84	58, 79	remulcld 8103	. . . . . . 7
85	78	absge0d 11495	. . . . . . . 8
86		absimle 11395	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9012	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11495	. . . . . . . 8
91	26	absnegd 11500	. . . . . . . . 9
92	59	renegcli 8334	. . . . . . . . . . . 12
93		0re 8072	. . . . . . . . . . . 12
94		pipos 15260	. . . . . . . . . . . . 13
95		lt0neg2 8542	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8175	. . . . . . . . . . 11
99	6	reim0d 11281	. . . . . . . . . . 11
100	98, 99	breqtrrid 4082	. . . . . . . . . 10
101	93, 59, 94	ltleii 8175	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4083	. . . . . . . . . 10
103		absle 11400	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 947	. . . . . . . . 9
106	91, 105	eqbrtrd 4066	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9013	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8196	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8196	. . . . 5
110		efle 15248	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9013	. . 3
114	51, 57, 63, 75, 113	letrd 8196	. 2
115	47, 114	eqbrtrd 4066	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1373

wcel 2176 class class class wbr 4044

cfv 5271 (class class class)co 5944

cc 7923

cr 7924

cc0 7925

caddc 7928

cmul 7930

clt 8107

cle 8108

cmin 8243

cneg 8244

crp 9775

cre 11151

cim 11152

cabs 11308

ce 11953

cpi 11958

clog 15328

ccxp 15329

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045 ax-pre-suploc 8046 ax-addf 8047 ax-mulf 8048

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-disj 4022 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-of 6158 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-oadd 6506 df-er 6620 df-map 6737 df-pm 6738 df-en 6828 df-dom 6829 df-fin 6830 df-sup 7086 df-inf 7087 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-9 9102 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-xneg 9894 df-xadd 9895 df-ioo 10014 df-ioc 10015 df-ico 10016 df-icc 10017 df-fz 10131 df-fzo 10265 df-seqfrec 10593 df-exp 10684 df-fac 10871 df-bc 10893 df-ihash 10921 df-shft 11126 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-clim 11590 df-sumdc 11665 df-ef 11959 df-e 11960 df-sin 11961 df-cos 11962 df-pi 11964 df-rest 13073 df-topgen 13092 df-psmet 14305 df-xmet 14306 df-met 14307 df-bl 14308 df-mopn 14309 df-top 14470 df-topon 14483 df-bases 14515 df-ntr 14568 df-cn 14660 df-cnp 14661 df-tx 14725 df-cncf 15043 df-limced 15128 df-dvap 15129 df-relog 15330 df-rpcxp 15331

This theorem is referenced by: (None)

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