sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12750

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 12688	. . . . 5 $/\$
2	1	nnsqcld 11081	. . . 4 $/\$
3	2	nncnd 9268	. . 3 $/\$
4	3	mulridd 8307	. 2 $/\$
5		nnsqcl 10995	. . . . . . 7
6	5	nnzd 9717	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10995	. . . . . . 7
9	8	nnzd 9717	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9613	. . . . . . . 8
12		nnz 9613	. . . . . . . 8
13		gcddvds 12684	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9717	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12745	. . . . . . 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 12745	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12740	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1282	. . . 4 $/\$
28		nncn 9262	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 9268	. . . . . 6 $/\$
31	1	nnap0d 9300	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 11073	. . . . 5 $/\$
33		nncn 9262	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 11073	. . . . 5 $/\$
36	32, 35	oveq12d 6076	. . . 4 $/\$
37		gcddiv 12740	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1277	. . . . . 6 $/\$
39	30, 31	dividapd 9077	. . . . . 6 $/\$
40	38, 39	eqtr3d 2269	. . . . 5 $/\$
41	1	nnne0d 9299	. . . . . . . . 9 $/\$
42		dvdsval2 12501	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1274	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 9261	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 9267	. . . . . . . 8 $/\$
48		nngt0 9279	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 9298	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 9226	. . . . . . 7 $/\$
52		elnnz 9604	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 12501	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1274	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 9261	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 9279	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 9226	. . . . . . 7 $/\$
62		elnnz 9604	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9416	. . . . . . 7
65		rppwr 12749	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1363	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 411	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2273	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 9299	. . . . . . . . 9
72	71	neneqd 2435	. . . . . . . 8
73	72	intnanrd 940	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 12683	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 9268	. . . 4 $/\$
78	2	nnap0d 9300	. . . 4 $/\$ #
79		ax-1cn 8236	. . . . 5
80		divmulap 8966	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1362	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1272	. . 3 $/\$
83	69, 82	mpbid 147	. 2 $/\$
84	4, 83	eqtr3d 2269	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2205

wne 2414 class class class wbr 4114 (class class class)co 6058

cc 8141

cr 8142

cc0 8143

c1 8144

cmul 8148

clt 8324 # cap 8872

cdiv 8963

cn 9254

c2 9305

cz 9594

cexp 10924

cdvds 12498

cgcd 12674

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-dvds 12499 df-gcd 12675

This theorem is referenced by: dvdssqlem 12751 nn0gcdsq 12922 pythagtriplem3 12990

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