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Theorem rpabscxpbnd 15114

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 15070	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5559	. . 3
6	1	relogcld 15058	. . . . . 6
7	6	recnd 8050	. . . . 5
8	2, 7	mulcld 8042	. . . 4
9		absef 11916	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11085	. . . . . . 7
12	7	recld 11085	. . . . . . 7
13	11, 12	remulcld 8052	. . . . . 6
14	13	recnd 8050	. . . . 5
15	2	imcld 11086	. . . . . . 7
16	7	imcld 11086	. . . . . . . 8
17	16	renegcld 8401	. . . . . . 7
18	15, 17	remulcld 8052	. . . . . 6
19	18	recnd 8050	. . . . 5
20		efadd 11821	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8052	. . . . . . . 8
23	22	recnd 8050	. . . . . . 7
24	14, 23	negsubd 8338	. . . . . 6
25	15	recnd 8050	. . . . . . . 8
26	16	recnd 8050	. . . . . . . 8
27	25, 26	mulneg2d 8433	. . . . . . 7
28	27	oveq2d 5935	. . . . . 6
29	2, 7	remuld 11110	. . . . . 6
30	24, 28, 29	3eqtr4d 2236	. . . . 5
31	30	fveq2d 5559	. . . 4
32	6	rered 11116	. . . . . . . . 9
33	1	rpred 9765	. . . . . . . . . . 11
34	1	rpge0d 9769	. . . . . . . . . . 11
35	33, 34	absidd 11314	. . . . . . . . . 10
36	35	fveq2d 5559	. . . . . . . . 9
37	32, 36	eqtr4d 2229	. . . . . . . 8
38	37	oveq2d 5935	. . . . . . 7
39	38	fveq2d 5559	. . . . . 6
40	35, 1	eqeltrd 2270	. . . . . . 7
41	11	recnd 8050	. . . . . . 7
42		rpcxpef 15070	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2229	. . . . 5
45	44	oveq1d 5934	. . . 4
46	21, 31, 45	3eqtr3d 2234	. . 3
47	5, 10, 46	3eqtrd 2230	. 2
48	40, 11	rpcxpcld 15107	. . . . 5
49	48	rpred 9765	. . . 4
50	18	reefcld 11815	. . . 4
51	49, 50	remulcld 8052	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9805	. . . . . 6
55	54, 11	rpcxpcld 15107	. . . . 5
56	55	rpred 9765	. . . 4
57	56, 50	remulcld 8052	. . 3
58	2	abscld 11328	. . . . . 6
59		pire 14962	. . . . . 6
60		remulcl 8002	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 11815	. . . 4
63	56, 62	remulcld 8052	. . 3
64	18	rpefcld 11832	. . . . 5
65	64	rpge0d 9769	. . . 4
66	1	rpcnd 9767	. . . . . . 7
67	1	rpap0d 9771	. . . . . . 7 #
68	66, 67	absrpclapd 11335	. . . . . 6
69	52, 68, 53	rpgecld 9805	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9762	. . . . . 6
72		rpcxple2 15093	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1249	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 8960	. . 3
76	55	rpge0d 9769	. . . 4
77	25	abscld 11328	. . . . . . 7
78	17	recnd 8050	. . . . . . . 8
79	78	abscld 11328	. . . . . . 7
80	77, 79	remulcld 8052	. . . . . 6
81	18	leabsd 11308	. . . . . . 7
82	25, 78	absmuld 11341	. . . . . . 7
83	81, 82	breqtrd 4056	. . . . . 6
84	58, 79	remulcld 8052	. . . . . . 7
85	78	absge0d 11331	. . . . . . . 8
86		absimle 11231	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 8960	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11331	. . . . . . . 8
91	26	absnegd 11336	. . . . . . . . 9
92	59	renegcli 8283	. . . . . . . . . . . 12
93		0re 8021	. . . . . . . . . . . 12
94		pipos 14964	. . . . . . . . . . . . 13
95		lt0neg2 8490	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8124	. . . . . . . . . . 11
99	6	reim0d 11117	. . . . . . . . . . 11
100	98, 99	breqtrrid 4068	. . . . . . . . . 10
101	93, 59, 94	ltleii 8124	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4069	. . . . . . . . . 10
103		absle 11236	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 946	. . . . . . . . 9
106	91, 105	eqbrtrd 4052	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 8961	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8145	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8145	. . . . 5
110		efle 14952	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 8961	. . 3
114	51, 57, 63, 75, 113	letrd 8145	. 2
115	47, 114	eqbrtrd 4052	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2164 class class class wbr 4030

cfv 5255 (class class class)co 5919

cc 7872

cr 7873

cc0 7874

caddc 7877

cmul 7879

clt 8056

cle 8057

cmin 8192

cneg 8193

crp 9722

cre 10987

cim 10988

cabs 11144

ce 11788

cpi 11793

clog 15032

ccxp 15033

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994 ax-pre-suploc 7995 ax-addf 7996 ax-mulf 7997

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-disj 4008 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-of 6132 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-map 6706 df-pm 6707 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-xneg 9841 df-xadd 9842 df-ioo 9961 df-ioc 9962 df-ico 9963 df-icc 9964 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-exp 10613 df-fac 10800 df-bc 10822 df-ihash 10850 df-shft 10962 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500 df-ef 11794 df-e 11795 df-sin 11796 df-cos 11797 df-pi 11799 df-rest 12855 df-topgen 12874 df-psmet 14042 df-xmet 14043 df-met 14044 df-bl 14045 df-mopn 14046 df-top 14177 df-topon 14190 df-bases 14222 df-ntr 14275 df-cn 14367 df-cnp 14368 df-tx 14432 df-cncf 14750 df-limced 14835 df-dvap 14836 df-relog 15034 df-rpcxp 15035

This theorem is referenced by: (None)

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