rpabscxpbnd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > rpabscxpbnd		Unicode version

Theorem rpabscxpbnd 15805

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 15759	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4
5	4	fveq2d 5674	. . 3
6	1	relogcld 15747	. . . . . 6
7	6	recnd 8302	. . . . 5
8	2, 7	mulcld 8294	. . . 4
9		absef 12456	. . . 4
10	8, 9	syl 14	. . 3
11	2	recld 11623	. . . . . . 7
12	7	recld 11623	. . . . . . 7
13	11, 12	remulcld 8304	. . . . . 6
14	13	recnd 8302	. . . . 5
15	2	imcld 11624	. . . . . . 7
16	7	imcld 11624	. . . . . . . 8
17	16	renegcld 8653	. . . . . . 7
18	15, 17	remulcld 8304	. . . . . 6
19	18	recnd 8302	. . . . 5
20		efadd 12361	. . . . 5 $/\$
21	14, 19, 20	syl2anc 411	. . . 4
22	15, 16	remulcld 8304	. . . . . . . 8
23	22	recnd 8302	. . . . . . 7
24	14, 23	negsubd 8590	. . . . . 6
25	15	recnd 8302	. . . . . . . 8
26	16	recnd 8302	. . . . . . . 8
27	25, 26	mulneg2d 8685	. . . . . . 7
28	27	oveq2d 6066	. . . . . 6
29	2, 7	remuld 11648	. . . . . 6
30	24, 28, 29	3eqtr4d 2275	. . . . 5
31	30	fveq2d 5674	. . . 4
32	6	rered 11654	. . . . . . . . 9
33	1	rpred 10029	. . . . . . . . . . 11
34	1	rpge0d 10033	. . . . . . . . . . 11
35	33, 34	absidd 11852	. . . . . . . . . 10
36	35	fveq2d 5674	. . . . . . . . 9
37	32, 36	eqtr4d 2268	. . . . . . . 8
38	37	oveq2d 6066	. . . . . . 7
39	38	fveq2d 5674	. . . . . 6
40	35, 1	eqeltrd 2309	. . . . . . 7
41	11	recnd 8302	. . . . . . 7
42		rpcxpef 15759	. . . . . . 7 $/\$
43	40, 41, 42	syl2anc 411	. . . . . 6
44	39, 43	eqtr4d 2268	. . . . 5
45	44	oveq1d 6065	. . . 4
46	21, 31, 45	3eqtr3d 2273	. . 3
47	5, 10, 46	3eqtrd 2269	. 2
48	40, 11	rpcxpcld 15798	. . . . 5
49	48	rpred 10029	. . . 4
50	18	reefcld 12355	. . . 4
51	49, 50	remulcld 8304	. . 3
52		abscxpbnd.4	. . . . . . 7
53		abscxpbnd.5	. . . . . . 7
54	52, 40, 53	rpgecld 10069	. . . . . 6
55	54, 11	rpcxpcld 15798	. . . . 5
56	55	rpred 10029	. . . 4
57	56, 50	remulcld 8304	. . 3
58	2	abscld 11866	. . . . . 6
59		pire 15651	. . . . . 6
60		remulcl 8255	. . . . . 6 $/\$
61	58, 59, 60	sylancl 413	. . . . 5
62	61	reefcld 12355	. . . 4
63	56, 62	remulcld 8304	. . 3
64	18	rpefcld 12372	. . . . 5
65	64	rpge0d 10033	. . . 4
66	1	rpcnd 10031	. . . . . . 7
67	1	rpap0d 10035	. . . . . . 7 #
68	66, 67	absrpclapd 11873	. . . . . 6
69	52, 68, 53	rpgecld 10069	. . . . . 6
70		rpabscxpbnd.3	. . . . . . 7
71	11, 70	elrpd 10026	. . . . . 6
72		rpcxple2 15783	. . . . . 6 $/\$ $/\$
73	68, 69, 71, 72	syl3anc 1274	. . . . 5
74	53, 73	mpbid 147	. . . 4
75	49, 56, 50, 65, 74	lemul1ad 9213	. . 3
76	55	rpge0d 10033	. . . 4
77	25	abscld 11866	. . . . . . 7
78	17	recnd 8302	. . . . . . . 8
79	78	abscld 11866	. . . . . . 7
80	77, 79	remulcld 8304	. . . . . 6
81	18	leabsd 11846	. . . . . . 7
82	25, 78	absmuld 11879	. . . . . . 7
83	81, 82	breqtrd 4135	. . . . . 6
84	58, 79	remulcld 8304	. . . . . . 7
85	78	absge0d 11869	. . . . . . . 8
86		absimle 11769	. . . . . . . . 9
87	2, 86	syl 14	. . . . . . . 8
88	77, 58, 79, 85, 87	lemul1ad 9213	. . . . . . 7
89	59	a1i 9	. . . . . . . 8
90	2	absge0d 11869	. . . . . . . 8
91	26	absnegd 11874	. . . . . . . . 9
92	59	renegcli 8535	. . . . . . . . . . . 12
93		0re 8274	. . . . . . . . . . . 12
94		pipos 15653	. . . . . . . . . . . . 13
95		lt0neg2 8743	. . . . . . . . . . . . . 14
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13
97	94, 96	mpbi 145	. . . . . . . . . . . 12
98	92, 93, 97	ltleii 8376	. . . . . . . . . . 11
99	6	reim0d 11655	. . . . . . . . . . 11
100	98, 99	breqtrrid 4147	. . . . . . . . . 10
101	93, 59, 94	ltleii 8376	. . . . . . . . . . 11
102	99, 101	eqbrtrdi 4148	. . . . . . . . . 10
103		absle 11774	. . . . . . . . . . 11 $/\$ $/\$
104	16, 59, 103	sylancl 413	. . . . . . . . . 10 $/\$
105	100, 102, 104	mpbir2and 953	. . . . . . . . 9
106	91, 105	eqbrtrd 4131	. . . . . . . 8
107	79, 89, 58, 90, 106	lemul2ad 9214	. . . . . . 7
108	80, 84, 61, 88, 107	letrd 8397	. . . . . 6
109	18, 80, 61, 83, 108	letrd 8397	. . . . 5
110		efle 15641	. . . . . 6 $/\$
111	18, 61, 110	syl2anc 411	. . . . 5
112	109, 111	mpbid 147	. . . 4
113	50, 62, 56, 76, 112	lemul2ad 9214	. . 3
114	51, 57, 63, 75, 113	letrd 8397	. 2
115	47, 114	eqbrtrd 4131	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2203 class class class wbr 4109

cfv 5352 (class class class)co 6050

cc 8125

cr 8126

cc0 8127

caddc 8130

cmul 8132

clt 8308

cle 8309

cmin 8444

cneg 8445

crp 9986

cre 11525

cim 11526

cabs 11682

ce 12328

cpi 12333

clog 15721

ccxp 15722

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247 ax-pre-suploc 8248 ax-addf 8249 ax-mulf 8250

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-disj 4086 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-of 6266 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-oadd 6651 df-er 6767 df-map 6884 df-pm 6885 df-en 6976 df-dom 6977 df-fin 6978 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-5 9299 df-6 9300 df-7 9301 df-8 9302 df-9 9303 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-xneg 10105 df-xadd 10106 df-ioo 10225 df-ioc 10226 df-ico 10227 df-icc 10228 df-fz 10343 df-fzo 10477 df-seqfrec 10810 df-exp 10901 df-fac 11088 df-bc 11110 df-ihash 11139 df-shft 11500 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-sumdc 12039 df-ef 12334 df-e 12335 df-sin 12336 df-cos 12337 df-pi 12339 df-rest 13454 df-topgen 13473 df-psmet 14691 df-xmet 14692 df-met 14693 df-bl 14694 df-mopn 14695 df-top 14863 df-topon 14876 df-bases 14908 df-ntr 14961 df-cn 15053 df-cnp 15054 df-tx 15118 df-cncf 15436 df-limced 15521 df-dvap 15522 df-relog 15723 df-rpcxp 15724

This theorem is referenced by: (None)

W3C validator