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Theorem rpabscxpbnd 15445

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 15399	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5582	. . 3
6	1	relogcld 15387	. . . . . 6
7	6	recnd 8103	. . . . 5
8	2, 7	mulcld 8095	. . . 4
9		absef 12114	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11282	. . . . . . 7
12	7	recld 11282	. . . . . . 7
13	11, 12	remulcld 8105	. . . . . 6
14	13	recnd 8103	. . . . 5
15	2	imcld 11283	. . . . . . 7
16	7	imcld 11283	. . . . . . . 8
17	16	renegcld 8454	. . . . . . 7
18	15, 17	remulcld 8105	. . . . . 6
19	18	recnd 8103	. . . . 5
20		efadd 12019	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8105	. . . . . . . 8
23	22	recnd 8103	. . . . . . 7
24	14, 23	negsubd 8391	. . . . . 6
25	15	recnd 8103	. . . . . . . 8
26	16	recnd 8103	. . . . . . . 8
27	25, 26	mulneg2d 8486	. . . . . . 7
28	27	oveq2d 5962	. . . . . 6
29	2, 7	remuld 11307	. . . . . 6
30	24, 28, 29	3eqtr4d 2248	. . . . 5
31	30	fveq2d 5582	. . . 4
32	6	rered 11313	. . . . . . . . 9
33	1	rpred 9820	. . . . . . . . . . 11
34	1	rpge0d 9824	. . . . . . . . . . 11
35	33, 34	absidd 11511	. . . . . . . . . 10
36	35	fveq2d 5582	. . . . . . . . 9
37	32, 36	eqtr4d 2241	. . . . . . . 8
38	37	oveq2d 5962	. . . . . . 7
39	38	fveq2d 5582	. . . . . 6
40	35, 1	eqeltrd 2282	. . . . . . 7
41	11	recnd 8103	. . . . . . 7
42		rpcxpef 15399	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2241	. . . . 5
45	44	oveq1d 5961	. . . 4
46	21, 31, 45	3eqtr3d 2246	. . 3
47	5, 10, 46	3eqtrd 2242	. 2
48	40, 11	rpcxpcld 15438	. . . . 5
49	48	rpred 9820	. . . 4
50	18	reefcld 12013	. . . 4
51	49, 50	remulcld 8105	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9860	. . . . . 6
55	54, 11	rpcxpcld 15438	. . . . 5
56	55	rpred 9820	. . . 4
57	56, 50	remulcld 8105	. . 3
58	2	abscld 11525	. . . . . 6
59		pire 15291	. . . . . 6
60		remulcl 8055	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12013	. . . 4
63	56, 62	remulcld 8105	. . 3
64	18	rpefcld 12030	. . . . 5
65	64	rpge0d 9824	. . . 4
66	1	rpcnd 9822	. . . . . . 7
67	1	rpap0d 9826	. . . . . . 7 #
68	66, 67	absrpclapd 11532	. . . . . 6
69	52, 68, 53	rpgecld 9860	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9817	. . . . . 6
72		rpcxple2 15423	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1250	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9014	. . 3
76	55	rpge0d 9824	. . . 4
77	25	abscld 11525	. . . . . . 7
78	17	recnd 8103	. . . . . . . 8
79	78	abscld 11525	. . . . . . 7
80	77, 79	remulcld 8105	. . . . . 6
81	18	leabsd 11505	. . . . . . 7
82	25, 78	absmuld 11538	. . . . . . 7
83	81, 82	breqtrd 4071	. . . . . 6
84	58, 79	remulcld 8105	. . . . . . 7
85	78	absge0d 11528	. . . . . . . 8
86		absimle 11428	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9014	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11528	. . . . . . . 8
91	26	absnegd 11533	. . . . . . . . 9
92	59	renegcli 8336	. . . . . . . . . . . 12
93		0re 8074	. . . . . . . . . . . 12
94		pipos 15293	. . . . . . . . . . . . 13
95		lt0neg2 8544	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8177	. . . . . . . . . . 11
99	6	reim0d 11314	. . . . . . . . . . 11
100	98, 99	breqtrrid 4083	. . . . . . . . . 10
101	93, 59, 94	ltleii 8177	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4084	. . . . . . . . . 10
103		absle 11433	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 947	. . . . . . . . 9
106	91, 105	eqbrtrd 4067	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9015	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8198	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8198	. . . . 5
110		efle 15281	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9015	. . 3
114	51, 57, 63, 75, 113	letrd 8198	. 2
115	47, 114	eqbrtrd 4067	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1373

wcel 2176 class class class wbr 4045

cfv 5272 (class class class)co 5946

cc 7925

cr 7926

cc0 7927

caddc 7930

cmul 7932

clt 8109

cle 8110

cmin 8245

cneg 8246

crp 9777

cre 11184

cim 11185

cabs 11341

ce 11986

cpi 11991

clog 15361

ccxp 15362

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 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045 ax-arch 8046 ax-caucvg 8047 ax-pre-suploc 8048 ax-addf 8049 ax-mulf 8050

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 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-isom 5281 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-of 6160 df-1st 6228 df-2nd 6229 df-recs 6393 df-irdg 6458 df-frec 6479 df-1o 6504 df-oadd 6508 df-er 6622 df-map 6739 df-pm 6740 df-en 6830 df-dom 6831 df-fin 6832 df-sup 7088 df-inf 7089 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-5 9100 df-6 9101 df-7 9102 df-8 9103 df-9 9104 df-n0 9298 df-z 9375 df-uz 9651 df-q 9743 df-rp 9778 df-xneg 9896 df-xadd 9897 df-ioo 10016 df-ioc 10017 df-ico 10018 df-icc 10019 df-fz 10133 df-fzo 10267 df-seqfrec 10595 df-exp 10686 df-fac 10873 df-bc 10895 df-ihash 10923 df-shft 11159 df-cj 11186 df-re 11187 df-im 11188 df-rsqrt 11342 df-abs 11343 df-clim 11623 df-sumdc 11698 df-ef 11992 df-e 11993 df-sin 11994 df-cos 11995 df-pi 11997 df-rest 13106 df-topgen 13125 df-psmet 14338 df-xmet 14339 df-met 14340 df-bl 14341 df-mopn 14342 df-top 14503 df-topon 14516 df-bases 14548 df-ntr 14601 df-cn 14693 df-cnp 14694 df-tx 14758 df-cncf 15076 df-limced 15161 df-dvap 15162 df-relog 15363 df-rpcxp 15364

This theorem is referenced by: (None)

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