sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 11997

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 11935	. . . . 5 $/\$
2	1	nnsqcld 10644	. . . 4 $/\$
3	2	nncnd 8906	. . 3 $/\$
4	3	mulid1d 7949	. 2 $/\$
5		nnsqcl 10559	. . . . . . 7
6	5	nnzd 9347	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10559	. . . . . . 7
9	8	nnzd 9347	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9245	. . . . . . . 8
12		nnz 9245	. . . . . . . 8
13		gcddvds 11931	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9347	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 11992	. . . . . . 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 11992	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 11987	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1246	. . . 4 $/\$
28		nncn 8900	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 8906	. . . . . 6 $/\$
31	1	nnap0d 8938	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10636	. . . . 5 $/\$
33		nncn 8900	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10636	. . . . 5 $/\$
36	32, 35	oveq12d 5883	. . . 4 $/\$
37		gcddiv 11987	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1241	. . . . . 6 $/\$
39	30, 31	dividapd 8716	. . . . . 6 $/\$
40	38, 39	eqtr3d 2210	. . . . 5 $/\$
41	1	nnne0d 8937	. . . . . . . . 9 $/\$
42		dvdsval2 11765	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1238	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 8899	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 8905	. . . . . . . 8 $/\$
48		nngt0 8917	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 8936	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8865	. . . . . . 7 $/\$
52		elnnz 9236	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 11765	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1238	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 8899	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 8917	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8865	. . . . . . 7 $/\$
62		elnnz 9236	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9053	. . . . . . 7
65		rppwr 11996	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1326	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 411	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2214	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8937	. . . . . . . . 9
72	71	neneqd 2366	. . . . . . . 8
73	72	intnanrd 932	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 11930	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 8906	. . . 4 $/\$
78	2	nnap0d 8938	. . . 4 $/\$ #
79		ax-1cn 7879	. . . . 5
80		divmulap 8605	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1325	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1236	. . 3 $/\$
83	69, 82	mpbid 147	. 2 $/\$
84	4, 83	eqtr3d 2210	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2146

wne 2345 class class class wbr 3998 (class class class)co 5865

cc 7784

cr 7785

cc0 7786

c1 7787

cmul 7791

clt 7966 # cap 8512

cdiv 8602

cn 8892

c2 8943

cz 9226

cexp 10489

cdvds 11762

cgcd 11910

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-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-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

This theorem is referenced by: dvdssqlem 11998 nn0gcdsq 12167 pythagtriplem3 12234

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