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Theorem rpabscxpbnd 15751

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 15705	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5652	. . 3
6	1	relogcld 15693	. . . . . 6
7	6	recnd 8267	. . . . 5
8	2, 7	mulcld 8259	. . . 4
9		absef 12411	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11578	. . . . . . 7
12	7	recld 11578	. . . . . . 7
13	11, 12	remulcld 8269	. . . . . 6
14	13	recnd 8267	. . . . 5
15	2	imcld 11579	. . . . . . 7
16	7	imcld 11579	. . . . . . . 8
17	16	renegcld 8618	. . . . . . 7
18	15, 17	remulcld 8269	. . . . . 6
19	18	recnd 8267	. . . . 5
20		efadd 12316	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8269	. . . . . . . 8
23	22	recnd 8267	. . . . . . 7
24	14, 23	negsubd 8555	. . . . . 6
25	15	recnd 8267	. . . . . . . 8
26	16	recnd 8267	. . . . . . . 8
27	25, 26	mulneg2d 8650	. . . . . . 7
28	27	oveq2d 6044	. . . . . 6
29	2, 7	remuld 11603	. . . . . 6
30	24, 28, 29	3eqtr4d 2274	. . . . 5
31	30	fveq2d 5652	. . . 4
32	6	rered 11609	. . . . . . . . 9
33	1	rpred 9992	. . . . . . . . . . 11
34	1	rpge0d 9996	. . . . . . . . . . 11
35	33, 34	absidd 11807	. . . . . . . . . 10
36	35	fveq2d 5652	. . . . . . . . 9
37	32, 36	eqtr4d 2267	. . . . . . . 8
38	37	oveq2d 6044	. . . . . . 7
39	38	fveq2d 5652	. . . . . 6
40	35, 1	eqeltrd 2308	. . . . . . 7
41	11	recnd 8267	. . . . . . 7
42		rpcxpef 15705	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2267	. . . . 5
45	44	oveq1d 6043	. . . 4
46	21, 31, 45	3eqtr3d 2272	. . 3
47	5, 10, 46	3eqtrd 2268	. 2
48	40, 11	rpcxpcld 15744	. . . . 5
49	48	rpred 9992	. . . 4
50	18	reefcld 12310	. . . 4
51	49, 50	remulcld 8269	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 10032	. . . . . 6
55	54, 11	rpcxpcld 15744	. . . . 5
56	55	rpred 9992	. . . 4
57	56, 50	remulcld 8269	. . 3
58	2	abscld 11821	. . . . . 6
59		pire 15597	. . . . . 6
60		remulcl 8220	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12310	. . . 4
63	56, 62	remulcld 8269	. . 3
64	18	rpefcld 12327	. . . . 5
65	64	rpge0d 9996	. . . 4
66	1	rpcnd 9994	. . . . . . 7
67	1	rpap0d 9998	. . . . . . 7 #
68	66, 67	absrpclapd 11828	. . . . . 6
69	52, 68, 53	rpgecld 10032	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9989	. . . . . 6
72		rpcxple2 15729	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1274	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9178	. . 3
76	55	rpge0d 9996	. . . 4
77	25	abscld 11821	. . . . . . 7
78	17	recnd 8267	. . . . . . . 8
79	78	abscld 11821	. . . . . . 7
80	77, 79	remulcld 8269	. . . . . 6
81	18	leabsd 11801	. . . . . . 7
82	25, 78	absmuld 11834	. . . . . . 7
83	81, 82	breqtrd 4119	. . . . . 6
84	58, 79	remulcld 8269	. . . . . . 7
85	78	absge0d 11824	. . . . . . . 8
86		absimle 11724	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9178	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11824	. . . . . . . 8
91	26	absnegd 11829	. . . . . . . . 9
92	59	renegcli 8500	. . . . . . . . . . . 12
93		0re 8239	. . . . . . . . . . . 12
94		pipos 15599	. . . . . . . . . . . . 13
95		lt0neg2 8708	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8341	. . . . . . . . . . 11
99	6	reim0d 11610	. . . . . . . . . . 11
100	98, 99	breqtrrid 4131	. . . . . . . . . 10
101	93, 59, 94	ltleii 8341	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4132	. . . . . . . . . 10
103		absle 11729	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 953	. . . . . . . . 9
106	91, 105	eqbrtrd 4115	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9179	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8362	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8362	. . . . 5
110		efle 15587	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9179	. . 3
114	51, 57, 63, 75, 113	letrd 8362	. 2
115	47, 114	eqbrtrd 4115	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2202 class class class wbr 4093

cfv 5333 (class class class)co 6028

cc 8090

cr 8091

cc0 8092

caddc 8095

cmul 8097

clt 8273

cle 8274

cmin 8409

cneg 8410

crp 9949

cre 11480

cim 11481

cabs 11637

ce 12283

cpi 12288

clog 15667

ccxp 15668

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212 ax-pre-suploc 8213 ax-addf 8214 ax-mulf 8215

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-disj 4070 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-of 6244 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-oadd 6629 df-er 6745 df-map 6862 df-pm 6863 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7243 df-inf 7244 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-5 9264 df-6 9265 df-7 9266 df-8 9267 df-9 9268 df-n0 9462 df-z 9541 df-uz 9817 df-q 9915 df-rp 9950 df-xneg 10068 df-xadd 10069 df-ioo 10188 df-ioc 10189 df-ico 10190 df-icc 10191 df-fz 10306 df-fzo 10440 df-seqfrec 10773 df-exp 10864 df-fac 11051 df-bc 11073 df-ihash 11101 df-shft 11455 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-clim 11919 df-sumdc 11994 df-ef 12289 df-e 12290 df-sin 12291 df-cos 12292 df-pi 12294 df-rest 13404 df-topgen 13423 df-psmet 14639 df-xmet 14640 df-met 14641 df-bl 14642 df-mopn 14643 df-top 14809 df-topon 14822 df-bases 14854 df-ntr 14907 df-cn 14999 df-cnp 15000 df-tx 15064 df-cncf 15382 df-limced 15467 df-dvap 15468 df-relog 15669 df-rpcxp 15670

This theorem is referenced by: (None)

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