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Theorem rpabscxpbnd 15931

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 15885	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5679	. . 3
6	1	relogcld 15873	. . . . . 6
7	6	recnd 8318	. . . . 5
8	2, 7	mulcld 8310	. . . 4
9		absef 12481	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11648	. . . . . . 7
12	7	recld 11648	. . . . . . 7
13	11, 12	remulcld 8320	. . . . . 6
14	13	recnd 8318	. . . . 5
15	2	imcld 11649	. . . . . . 7
16	7	imcld 11649	. . . . . . . 8
17	16	renegcld 8670	. . . . . . 7
18	15, 17	remulcld 8320	. . . . . 6
19	18	recnd 8318	. . . . 5
20		efadd 12386	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8320	. . . . . . . 8
23	22	recnd 8318	. . . . . . 7
24	14, 23	negsubd 8606	. . . . . 6
25	15	recnd 8318	. . . . . . . 8
26	16	recnd 8318	. . . . . . . 8
27	25, 26	mulneg2d 8702	. . . . . . 7
28	27	oveq2d 6074	. . . . . 6
29	2, 7	remuld 11673	. . . . . 6
30	24, 28, 29	3eqtr4d 2277	. . . . 5
31	30	fveq2d 5679	. . . 4
32	6	rered 11679	. . . . . . . . 9
33	1	rpred 10047	. . . . . . . . . . 11
34	1	rpge0d 10051	. . . . . . . . . . 11
35	33, 34	absidd 11877	. . . . . . . . . 10
36	35	fveq2d 5679	. . . . . . . . 9
37	32, 36	eqtr4d 2270	. . . . . . . 8
38	37	oveq2d 6074	. . . . . . 7
39	38	fveq2d 5679	. . . . . 6
40	35, 1	eqeltrd 2311	. . . . . . 7
41	11	recnd 8318	. . . . . . 7
42		rpcxpef 15885	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2270	. . . . 5
45	44	oveq1d 6073	. . . 4
46	21, 31, 45	3eqtr3d 2275	. . 3
47	5, 10, 46	3eqtrd 2271	. 2
48	40, 11	rpcxpcld 15924	. . . . 5
49	48	rpred 10047	. . . 4
50	18	reefcld 12380	. . . 4
51	49, 50	remulcld 8320	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 10087	. . . . . 6
55	54, 11	rpcxpcld 15924	. . . . 5
56	55	rpred 10047	. . . 4
57	56, 50	remulcld 8320	. . 3
58	2	abscld 11891	. . . . . 6
59		pire 15777	. . . . . 6
60		remulcl 8271	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12380	. . . 4
63	56, 62	remulcld 8320	. . 3
64	18	rpefcld 12397	. . . . 5
65	64	rpge0d 10051	. . . 4
66	1	rpcnd 10049	. . . . . . 7
67	1	rpap0d 10053	. . . . . . 7 #
68	66, 67	absrpclapd 11898	. . . . . 6
69	52, 68, 53	rpgecld 10087	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 10044	. . . . . 6
72		rpcxple2 15909	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1274	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9230	. . 3
76	55	rpge0d 10051	. . . 4
77	25	abscld 11891	. . . . . . 7
78	17	recnd 8318	. . . . . . . 8
79	78	abscld 11891	. . . . . . 7
80	77, 79	remulcld 8320	. . . . . 6
81	18	leabsd 11871	. . . . . . 7
82	25, 78	absmuld 11904	. . . . . . 7
83	81, 82	breqtrd 4140	. . . . . 6
84	58, 79	remulcld 8320	. . . . . . 7
85	78	absge0d 11894	. . . . . . . 8
86		absimle 11794	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9230	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11894	. . . . . . . 8
91	26	absnegd 11899	. . . . . . . . 9
92	59	renegcli 8551	. . . . . . . . . . . 12
93		0re 8290	. . . . . . . . . . . 12
94		pipos 15779	. . . . . . . . . . . . 13
95		lt0neg2 8760	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8392	. . . . . . . . . . 11
99	6	reim0d 11680	. . . . . . . . . . 11
100	98, 99	breqtrrid 4152	. . . . . . . . . 10
101	93, 59, 94	ltleii 8392	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4153	. . . . . . . . . 10
103		absle 11799	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 953	. . . . . . . . 9
106	91, 105	eqbrtrd 4136	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9231	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8413	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8413	. . . . 5
110		efle 15767	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9231	. . 3
114	51, 57, 63, 75, 113	letrd 8413	. 2
115	47, 114	eqbrtrd 4136	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2205 class class class wbr 4114

cfv 5357 (class class class)co 6058

cc 8141

cr 8142

cc0 8143

caddc 8146

cmul 8148

clt 8324

cle 8325

cmin 8460

cneg 8461

crp 10004

cre 11550

cim 11551

cabs 11707

ce 12353

cpi 12358

clog 15847

ccxp 15848

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-pre-suploc 8264 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ioo 10244 df-ioc 10245 df-ico 10246 df-icc 10247 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-fac 11113 df-bc 11135 df-ihash 11164 df-shft 11525 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064 df-ef 12359 df-e 12360 df-sin 12361 df-cos 12362 df-pi 12364 df-rest 13538 df-topgen 13557 df-psmet 14817 df-xmet 14818 df-met 14819 df-bl 14820 df-mopn 14821 df-top 14989 df-topon 15002 df-bases 15034 df-ntr 15087 df-cn 15179 df-cnp 15180 df-tx 15244 df-cncf 15562 df-limced 15647 df-dvap 15648 df-relog 15849 df-rpcxp 15850

This theorem is referenced by: (None)

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