2sqpwodd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 2sqpwodd		Unicode version

Theorem 2sqpwodd 12143

Description: The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)

**Hypotheses**
Ref	Expression
oddpwdc.j
oddpwdc.f

**Assertion**
Ref	Expression
2sqpwodd

Distinct variable groups:

Allowed substitution hint:

(

)

**Proof of Theorem 2sqpwodd**
Step	Hyp	Ref	Expression
1		oddpwdc.j	. . . . . . . . 9
2		oddpwdc.f	. . . . . . . . 9
3	1, 2	oddpwdc 12141	. . . . . . . 8
4		f1ocnv 5466	. . . . . . . 8
5		f1of 5453	. . . . . . . 8
6	3, 4, 5	mp2b 8	. . . . . . 7
7	6	ffvelcdmi 5642	. . . . . 6
8		xp2nd 6157	. . . . . 6
9	7, 8	syl 14	. . . . 5
10	9	nn0zd 9346	. . . 4
11		2nn 9053	. . . . . 6
12	11	a1i 9	. . . . 5
13	12	nnzd 9347	. . . 4
14	10, 13	zmulcld 9354	. . 3
15		dvdsmul2 11789	. . . 4 $/\$
16	10, 13, 15	syl2anc 411	. . 3
17		oddp1even 11848	. . . . 5
18	17	biimprd 158	. . . 4
19	18	con2d 624	. . 3
20	14, 16, 19	sylc 62	. 2
21		xp1st 6156	. . . . . . . . . . 11
22	7, 21	syl 14	. . . . . . . . . 10
23		breq2 4002	. . . . . . . . . . . . 13
24	23	notbid 667	. . . . . . . . . . . 12
25	24, 1	elrab2 2894	. . . . . . . . . . 11 $/\$
26	25	simplbi 274	. . . . . . . . . 10
27	22, 26	syl 14	. . . . . . . . 9
28	27	nnsqcld 10644	. . . . . . . 8
29	25	simprbi 275	. . . . . . . . . . 11
30	22, 29	syl 14	. . . . . . . . . 10
31		2prm 12094	. . . . . . . . . . 11
32	27	nnzd 9347	. . . . . . . . . . 11
33		euclemma 12113	. . . . . . . . . . . 12 $/\$ $/\$ $\/$
34		oridm 757	. . . . . . . . . . . 12 $\/$
35	33, 34	bitrdi 196	. . . . . . . . . . 11 $/\$ $/\$
36	31, 32, 32, 35	mp3an2i 1342	. . . . . . . . . 10
37	30, 36	mtbird 673	. . . . . . . . 9
38	27	nncnd 8906	. . . . . . . . . . 11
39	38	sqvald 10620	. . . . . . . . . 10
40	39	breq2d 4010	. . . . . . . . 9
41	37, 40	mtbird 673	. . . . . . . 8
42		breq2 4002	. . . . . . . . . 10
43	42	notbid 667	. . . . . . . . 9
44	43, 1	elrab2 2894	. . . . . . . 8 $/\$
45	28, 41, 44	sylanbrc 417	. . . . . . 7
46	12	nnnn0d 9202	. . . . . . . . 9
47	9, 46	nn0mulcld 9207	. . . . . . . 8
48		peano2nn0 9189	. . . . . . . 8
49	47, 48	syl 14	. . . . . . 7
50		opelxp 4650	. . . . . . 7 $/\$
51	45, 49, 50	sylanbrc 417	. . . . . 6
52	12	nncnd 8906	. . . . . . . . . . 11
53	52, 47	expp1d 10624	. . . . . . . . . 10
54	52, 47	expcld 10623	. . . . . . . . . . 11
55	54, 52	mulcomd 7953	. . . . . . . . . 10
56	52, 46, 9	expmuld 10626	. . . . . . . . . . 11
57	56	oveq2d 5881	. . . . . . . . . 10
58	53, 55, 57	3eqtrd 2212	. . . . . . . . 9
59	58	oveq1d 5880	. . . . . . . 8
60	12, 49	nnexpcld 10645	. . . . . . . . . 10
61	60, 28	nnmulcld 8941	. . . . . . . . 9
62		oveq2 5873	. . . . . . . . . 10
63		oveq2 5873	. . . . . . . . . . 11
64	63	oveq1d 5880	. . . . . . . . . 10
65	62, 64, 2	ovmpog 5999	. . . . . . . . 9 $/\$ $/\$
66	45, 49, 61, 65	syl3anc 1238	. . . . . . . 8
67		f1ocnvfv2 5769	. . . . . . . . . . . . . . . 16 $/\$
68	3, 67	mpan 424	. . . . . . . . . . . . . . 15
69		1st2nd2 6166	. . . . . . . . . . . . . . . . 17
70	7, 69	syl 14	. . . . . . . . . . . . . . . 16
71	70	fveq2d 5511	. . . . . . . . . . . . . . 15
72	68, 71	eqtr3d 2210	. . . . . . . . . . . . . 14
73		df-ov 5868	. . . . . . . . . . . . . 14
74	72, 73	eqtr4di 2226	. . . . . . . . . . . . 13
75	12, 9	nnexpcld 10645	. . . . . . . . . . . . . . 15
76	75, 27	nnmulcld 8941	. . . . . . . . . . . . . 14
77		oveq2 5873	. . . . . . . . . . . . . . 15
78		oveq2 5873	. . . . . . . . . . . . . . . 16
79	78	oveq1d 5880	. . . . . . . . . . . . . . 15
80	77, 79, 2	ovmpog 5999	. . . . . . . . . . . . . 14 $/\$ $/\$
81	22, 9, 76, 80	syl3anc 1238	. . . . . . . . . . . . 13
82	74, 81	eqtrd 2208	. . . . . . . . . . . 12
83	82	oveq1d 5880	. . . . . . . . . . 11
84	75	nncnd 8906	. . . . . . . . . . . 12
85	84, 38	sqmuld 10635	. . . . . . . . . . 11
86	83, 85	eqtrd 2208	. . . . . . . . . 10
87	86	oveq2d 5881	. . . . . . . . 9
88	56, 54	eqeltrrd 2253	. . . . . . . . . 10
89	28	nncnd 8906	. . . . . . . . . 10
90	52, 88, 89	mulassd 7955	. . . . . . . . 9
91	87, 90	eqtr4d 2211	. . . . . . . 8
92	59, 66, 91	3eqtr4rd 2219	. . . . . . 7
93		df-ov 5868	. . . . . . 7
94	92, 93	eqtr2di 2225	. . . . . 6
95		f1ocnvfv 5770	. . . . . . 7 $/\$
96	3, 95	mpan 424	. . . . . 6
97	51, 94, 96	sylc 62	. . . . 5
98	97	fveq2d 5511	. . . 4
99		op2ndg 6142	. . . . 5 $/\$
100	45, 49, 99	syl2anc 411	. . . 4
101	98, 100	eqtrd 2208	. . 3
102	101	breq2d 4010	. 2
103	20, 102	mtbird 673	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 105 $\/$ wo 708 $/\$ w3a 978

wceq 1353

wcel 2146

crab 2457

cop 3592 class class class wbr 3998

cxp 4618

ccnv 4619

wf 5204

wf1o 5207

cfv 5208 (class class class)co 5865

cmpo 5867

c1st 6129

c2nd 6130

cc 7784

c1 7787

caddc 7789

cmul 7791

cn 8892

c2 8943

cn0 9149

cz 9226

cexp 10489

cdvds 11762

cprime 12074

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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906

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-xor 1376 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-1o 6407 df-2o 6408 df-er 6525 df-en 6731 df-sup 6973 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-fz 9980 df-fzo 10113 df-fl 10240 df-mod 10293 df-seqfrec 10416 df-exp 10490 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-dvds 11763 df-gcd 11911 df-prm 12075

This theorem is referenced by: sqne2sq 12144

W3C validator