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Theorem rpabscxpbnd 14444

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 14400	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5521	. . 3
6	1	relogcld 14388	. . . . . 6
7	6	recnd 7988	. . . . 5
8	2, 7	mulcld 7980	. . . 4
9		absef 11779	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 10949	. . . . . . 7
12	7	recld 10949	. . . . . . 7
13	11, 12	remulcld 7990	. . . . . 6
14	13	recnd 7988	. . . . 5
15	2	imcld 10950	. . . . . . 7
16	7	imcld 10950	. . . . . . . 8
17	16	renegcld 8339	. . . . . . 7
18	15, 17	remulcld 7990	. . . . . 6
19	18	recnd 7988	. . . . 5
20		efadd 11685	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 7990	. . . . . . . 8
23	22	recnd 7988	. . . . . . 7
24	14, 23	negsubd 8276	. . . . . 6
25	15	recnd 7988	. . . . . . . 8
26	16	recnd 7988	. . . . . . . 8
27	25, 26	mulneg2d 8371	. . . . . . 7
28	27	oveq2d 5893	. . . . . 6
29	2, 7	remuld 10974	. . . . . 6
30	24, 28, 29	3eqtr4d 2220	. . . . 5
31	30	fveq2d 5521	. . . 4
32	6	rered 10980	. . . . . . . . 9
33	1	rpred 9698	. . . . . . . . . . 11
34	1	rpge0d 9702	. . . . . . . . . . 11
35	33, 34	absidd 11178	. . . . . . . . . 10
36	35	fveq2d 5521	. . . . . . . . 9
37	32, 36	eqtr4d 2213	. . . . . . . 8
38	37	oveq2d 5893	. . . . . . 7
39	38	fveq2d 5521	. . . . . 6
40	35, 1	eqeltrd 2254	. . . . . . 7
41	11	recnd 7988	. . . . . . 7
42		rpcxpef 14400	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2213	. . . . 5
45	44	oveq1d 5892	. . . 4
46	21, 31, 45	3eqtr3d 2218	. . 3
47	5, 10, 46	3eqtrd 2214	. 2
48	40, 11	rpcxpcld 14437	. . . . 5
49	48	rpred 9698	. . . 4
50	18	reefcld 11679	. . . 4
51	49, 50	remulcld 7990	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 9738	. . . . . 6
55	54, 11	rpcxpcld 14437	. . . . 5
56	55	rpred 9698	. . . 4
57	56, 50	remulcld 7990	. . 3
58	2	abscld 11192	. . . . . 6
59		pire 14292	. . . . . 6
60		remulcl 7941	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 11679	. . . 4
63	56, 62	remulcld 7990	. . 3
64	18	rpefcld 11696	. . . . 5
65	64	rpge0d 9702	. . . 4
66	1	rpcnd 9700	. . . . . . 7
67	1	rpap0d 9704	. . . . . . 7 #
68	66, 67	absrpclapd 11199	. . . . . 6
69	52, 68, 53	rpgecld 9738	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 9695	. . . . . 6
72		rpcxple2 14423	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1238	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 8898	. . 3
76	55	rpge0d 9702	. . . 4
77	25	abscld 11192	. . . . . . 7
78	17	recnd 7988	. . . . . . . 8
79	78	abscld 11192	. . . . . . 7
80	77, 79	remulcld 7990	. . . . . 6
81	18	leabsd 11172	. . . . . . 7
82	25, 78	absmuld 11205	. . . . . . 7
83	81, 82	breqtrd 4031	. . . . . 6
84	58, 79	remulcld 7990	. . . . . . 7
85	78	absge0d 11195	. . . . . . . 8
86		absimle 11095	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 8898	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11195	. . . . . . . 8
91	26	absnegd 11200	. . . . . . . . 9
92	59	renegcli 8221	. . . . . . . . . . . 12
93		0re 7959	. . . . . . . . . . . 12
94		pipos 14294	. . . . . . . . . . . . 13
95		lt0neg2 8428	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8062	. . . . . . . . . . 11
99	6	reim0d 10981	. . . . . . . . . . 11
100	98, 99	breqtrrid 4043	. . . . . . . . . 10
101	93, 59, 94	ltleii 8062	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4044	. . . . . . . . . 10
103		absle 11100	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 944	. . . . . . . . 9
106	91, 105	eqbrtrd 4027	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 8899	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8083	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8083	. . . . 5
110		efle 14282	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 8899	. . 3
114	51, 57, 63, 75, 113	letrd 8083	. 2
115	47, 114	eqbrtrd 4027	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148 class class class wbr 4005

cfv 5218 (class class class)co 5877

cc 7811

cr 7812

cc0 7813

caddc 7816

cmul 7818

clt 7994

cle 7995

cmin 8130

cneg 8131

crp 9655

cre 10851

cim 10852

cabs 11008

ce 11652

cpi 11657

clog 14362

ccxp 14363

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-pre-suploc 7934 ax-addf 7935 ax-mulf 7936

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-map 6652 df-pm 6653 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-7 8985 df-8 8986 df-9 8987 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-ioo 9894 df-ioc 9895 df-ico 9896 df-icc 9897 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-shft 10826 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364 df-ef 11658 df-e 11659 df-sin 11660 df-cos 11661 df-pi 11663 df-rest 12695 df-topgen 12714 df-psmet 13532 df-xmet 13533 df-met 13534 df-bl 13535 df-mopn 13536 df-top 13583 df-topon 13596 df-bases 13628 df-ntr 13681 df-cn 13773 df-cnp 13774 df-tx 13838 df-cncf 14143 df-limced 14210 df-dvap 14211 df-relog 14364 df-rpcxp 14365

This theorem is referenced by: (None)

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