sqgcd - Intuitionistic Logic Explorer

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

Theorem sqgcd 12062

Description: Square distributes over gcd. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
sqgcd	$/\$

**Proof of Theorem sqgcd**
Step	Hyp	Ref	Expression
1		gcdnncl 12000	. . . . 5 $/\$
2	1	nnsqcld 10706	. . . 4 $/\$
3	2	nncnd 8963	. . 3 $/\$
4	3	mulridd 8004	. 2 $/\$
5		nnsqcl 10621	. . . . . . 7
6	5	nnzd 9404	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10621	. . . . . . 7
9	8	nnzd 9404	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9302	. . . . . . . 8
12		nnz 9302	. . . . . . . 8
13		gcddvds 11996	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9404	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12057	. . . . . . 7 $/\$
19	16, 17, 18	syl2anc 411	. . . . . 6 $/\$
20	15, 19	mpd 13	. . . . 5 $/\$
21	14	simprd 114	. . . . . 6 $/\$
22	12	adantl 277	. . . . . . 7 $/\$
23		dvdssqim 12057	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12052	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1257	. . . 4 $/\$
28		nncn 8957	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 8963	. . . . . 6 $/\$
31	1	nnap0d 8995	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10698	. . . . 5 $/\$
33		nncn 8957	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10698	. . . . 5 $/\$
36	32, 35	oveq12d 5914	. . . 4 $/\$
37		gcddiv 12052	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1252	. . . . . 6 $/\$
39	30, 31	dividapd 8773	. . . . . 6 $/\$
40	38, 39	eqtr3d 2224	. . . . 5 $/\$
41	1	nnne0d 8994	. . . . . . . . 9 $/\$
42		dvdsval2 11829	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1249	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 8956	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 8962	. . . . . . . 8 $/\$
48		nngt0 8974	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 8993	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8922	. . . . . . 7 $/\$
52		elnnz 9293	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 11829	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1249	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 8956	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 8974	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8922	. . . . . . 7 $/\$
62		elnnz 9293	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9110	. . . . . . 7
65		rppwr 12061	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1337	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 411	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2228	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8994	. . . . . . . . 9
72	71	neneqd 2381	. . . . . . . 8
73	72	intnanrd 933	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 11995	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 8963	. . . 4 $/\$
78	2	nnap0d 8995	. . . 4 $/\$ #
79		ax-1cn 7934	. . . . 5
80		divmulap 8662	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1336	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1247	. . 3 $/\$
83	69, 82	mpbid 147	. 2 $/\$
84	4, 83	eqtr3d 2224	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2160

wne 2360 class class class wbr 4018 (class class class)co 5896

cc 7839

cr 7840

cc0 7841

c1 7842

cmul 7846

clt 8022 # cap 8568

cdiv 8659

cn 8949

c2 9000

cz 9283

cexp 10550

cdvds 11826

cgcd 11975

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-frec 6416 df-sup 7013 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-fz 10039 df-fzo 10173 df-fl 10301 df-mod 10354 df-seqfrec 10477 df-exp 10551 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-dvds 11827 df-gcd 11976

This theorem is referenced by: dvdssqlem 12063 nn0gcdsq 12232 pythagtriplem3 12299

W3C validator