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Theorem rpabscxpbnd 15176

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 15130	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5562	. . 3
6	1	relogcld 15118	. . . . . 6
7	6	recnd 8055	. . . . 5
8	2, 7	mulcld 8047	. . . 4
9		absef 11935	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11103	. . . . . . 7
12	7	recld 11103	. . . . . . 7
13	11, 12	remulcld 8057	. . . . . 6
14	13	recnd 8055	. . . . 5
15	2	imcld 11104	. . . . . . 7
16	7	imcld 11104	. . . . . . . 8
17	16	renegcld 8406	. . . . . . 7
18	15, 17	remulcld 8057	. . . . . 6
19	18	recnd 8055	. . . . 5
20		efadd 11840	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8057	. . . . . . . 8
23	22	recnd 8055	. . . . . . 7
24	14, 23	negsubd 8343	. . . . . 6
25	15	recnd 8055	. . . . . . . 8
26	16	recnd 8055	. . . . . . . 8
27	25, 26	mulneg2d 8438	. . . . . . 7
28	27	oveq2d 5938	. . . . . 6
29	2, 7	remuld 11128	. . . . . 6
30	24, 28, 29	3eqtr4d 2239	. . . . 5
31	30	fveq2d 5562	. . . 4
32	6	rered 11134	. . . . . . . . 9
33	1	rpred 9771	. . . . . . . . . . 11
34	1	rpge0d 9775	. . . . . . . . . . 11
35	33, 34	absidd 11332	. . . . . . . . . 10
36	35	fveq2d 5562	. . . . . . . . 9
37	32, 36	eqtr4d 2232	. . . . . . . 8
38	37	oveq2d 5938	. . . . . . 7
39	38	fveq2d 5562	. . . . . 6
40	35, 1	eqeltrd 2273	. . . . . . 7
41	11	recnd 8055	. . . . . . 7
42		rpcxpef 15130	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2232	. . . . 5
45	44	oveq1d 5937	. . . 4
46	21, 31, 45	3eqtr3d 2237	. . 3
47	5, 10, 46	3eqtrd 2233	. 2
48	40, 11	rpcxpcld 15169	. . . . 5
49	48	rpred 9771	. . . 4
50	18	reefcld 11834	. . . 4
51	49, 50	remulcld 8057	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9811	. . . . . 6
55	54, 11	rpcxpcld 15169	. . . . 5
56	55	rpred 9771	. . . 4
57	56, 50	remulcld 8057	. . 3
58	2	abscld 11346	. . . . . 6
59		pire 15022	. . . . . 6
60		remulcl 8007	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 11834	. . . 4
63	56, 62	remulcld 8057	. . 3
64	18	rpefcld 11851	. . . . 5
65	64	rpge0d 9775	. . . 4
66	1	rpcnd 9773	. . . . . . 7
67	1	rpap0d 9777	. . . . . . 7 #
68	66, 67	absrpclapd 11353	. . . . . 6
69	52, 68, 53	rpgecld 9811	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9768	. . . . . 6
72		rpcxple2 15154	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1249	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 8966	. . 3
76	55	rpge0d 9775	. . . 4
77	25	abscld 11346	. . . . . . 7
78	17	recnd 8055	. . . . . . . 8
79	78	abscld 11346	. . . . . . 7
80	77, 79	remulcld 8057	. . . . . 6
81	18	leabsd 11326	. . . . . . 7
82	25, 78	absmuld 11359	. . . . . . 7
83	81, 82	breqtrd 4059	. . . . . 6
84	58, 79	remulcld 8057	. . . . . . 7
85	78	absge0d 11349	. . . . . . . 8
86		absimle 11249	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 8966	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11349	. . . . . . . 8
91	26	absnegd 11354	. . . . . . . . 9
92	59	renegcli 8288	. . . . . . . . . . . 12
93		0re 8026	. . . . . . . . . . . 12
94		pipos 15024	. . . . . . . . . . . . 13
95		lt0neg2 8496	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8129	. . . . . . . . . . 11
99	6	reim0d 11135	. . . . . . . . . . 11
100	98, 99	breqtrrid 4071	. . . . . . . . . 10
101	93, 59, 94	ltleii 8129	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4072	. . . . . . . . . 10
103		absle 11254	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 946	. . . . . . . . 9
106	91, 105	eqbrtrd 4055	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 8967	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8150	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8150	. . . . 5
110		efle 15012	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 8967	. . 3
114	51, 57, 63, 75, 113	letrd 8150	. 2
115	47, 114	eqbrtrd 4055	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2167 class class class wbr 4033

cfv 5258 (class class class)co 5922

cc 7877

cr 7878

cc0 7879

caddc 7882

cmul 7884

clt 8061

cle 8062

cmin 8197

cneg 8198

crp 9728

cre 11005

cim 11006

cabs 11162

ce 11807

cpi 11812

clog 15092

ccxp 15093

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999 ax-pre-suploc 8000 ax-addf 8001 ax-mulf 8002

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-disj 4011 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-of 6135 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-map 6709 df-pm 6710 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-5 9052 df-6 9053 df-7 9054 df-8 9055 df-9 9056 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-xneg 9847 df-xadd 9848 df-ioo 9967 df-ioc 9968 df-ico 9969 df-icc 9970 df-fz 10084 df-fzo 10218 df-seqfrec 10540 df-exp 10631 df-fac 10818 df-bc 10840 df-ihash 10868 df-shft 10980 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-sumdc 11519 df-ef 11813 df-e 11814 df-sin 11815 df-cos 11816 df-pi 11818 df-rest 12912 df-topgen 12931 df-psmet 14099 df-xmet 14100 df-met 14101 df-bl 14102 df-mopn 14103 df-top 14234 df-topon 14247 df-bases 14279 df-ntr 14332 df-cn 14424 df-cnp 14425 df-tx 14489 df-cncf 14807 df-limced 14892 df-dvap 14893 df-relog 15094 df-rpcxp 15095

This theorem is referenced by: (None)

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