sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12196

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 12134	. . . . 5 $/\$
2	1	nnsqcld 10786	. . . 4 $/\$
3	2	nncnd 9004	. . 3 $/\$
4	3	mulridd 8043	. 2 $/\$
5		nnsqcl 10701	. . . . . . 7
6	5	nnzd 9447	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10701	. . . . . . 7
9	8	nnzd 9447	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9345	. . . . . . . 8
12		nnz 9345	. . . . . . . 8
13		gcddvds 12130	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9447	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12191	. . . . . . 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 12191	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12186	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1257	. . . 4 $/\$
28		nncn 8998	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 9004	. . . . . 6 $/\$
31	1	nnap0d 9036	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10778	. . . . 5 $/\$
33		nncn 8998	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10778	. . . . 5 $/\$
36	32, 35	oveq12d 5940	. . . 4 $/\$
37		gcddiv 12186	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1252	. . . . . 6 $/\$
39	30, 31	dividapd 8813	. . . . . 6 $/\$
40	38, 39	eqtr3d 2231	. . . . 5 $/\$
41	1	nnne0d 9035	. . . . . . . . 9 $/\$
42		dvdsval2 11955	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1249	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 8997	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 9003	. . . . . . . 8 $/\$
48		nngt0 9015	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 9034	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8962	. . . . . . 7 $/\$
52		elnnz 9336	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 11955	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1249	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 8997	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 9015	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8962	. . . . . . 7 $/\$
62		elnnz 9336	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9152	. . . . . . 7
65		rppwr 12195	. . . . . . 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 2235	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 9035	. . . . . . . . 9
72	71	neneqd 2388	. . . . . . . 8
73	72	intnanrd 933	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 12129	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 9004	. . . 4 $/\$
78	2	nnap0d 9036	. . . 4 $/\$ #
79		ax-1cn 7972	. . . . 5
80		divmulap 8702	. . . . 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 2231	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2167

wne 2367 class class class wbr 4033 (class class class)co 5922

cc 7877

cr 7878

cc0 7879

c1 7880

cmul 7884

clt 8061 # cap 8608

cdiv 8699

cn 8990

c2 9041

cz 9326

cexp 10630

cdvds 11952

cgcd 12120

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-sup 7050 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121

This theorem is referenced by: dvdssqlem 12197 nn0gcdsq 12368 pythagtriplem3 12436

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