sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12166

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 12104	. . . . 5 $/\$
2	1	nnsqcld 10765	. . . 4 $/\$
3	2	nncnd 8996	. . 3 $/\$
4	3	mulridd 8036	. 2 $/\$
5		nnsqcl 10680	. . . . . . 7
6	5	nnzd 9438	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10680	. . . . . . 7
9	8	nnzd 9438	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9336	. . . . . . . 8
12		nnz 9336	. . . . . . . 8
13		gcddvds 12100	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9438	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12161	. . . . . . 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 12161	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12156	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1257	. . . 4 $/\$
28		nncn 8990	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 8996	. . . . . 6 $/\$
31	1	nnap0d 9028	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10757	. . . . 5 $/\$
33		nncn 8990	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10757	. . . . 5 $/\$
36	32, 35	oveq12d 5936	. . . 4 $/\$
37		gcddiv 12156	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1252	. . . . . 6 $/\$
39	30, 31	dividapd 8805	. . . . . 6 $/\$
40	38, 39	eqtr3d 2228	. . . . 5 $/\$
41	1	nnne0d 9027	. . . . . . . . 9 $/\$
42		dvdsval2 11933	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1249	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 8989	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 8995	. . . . . . . 8 $/\$
48		nngt0 9007	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 9026	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8954	. . . . . . 7 $/\$
52		elnnz 9327	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 11933	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1249	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 8989	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 9007	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8954	. . . . . . 7 $/\$
62		elnnz 9327	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9143	. . . . . . 7
65		rppwr 12165	. . . . . . 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 2232	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 9027	. . . . . . . . 9
72	71	neneqd 2385	. . . . . . . 8
73	72	intnanrd 933	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 12099	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 8996	. . . 4 $/\$
78	2	nnap0d 9028	. . . 4 $/\$ #
79		ax-1cn 7965	. . . . 5
80		divmulap 8694	. . . . 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 2228	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2164

wne 2364 class class class wbr 4029 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

c1 7873

cmul 7877

clt 8054 # cap 8600

cdiv 8691

cn 8982

c2 9033

cz 9317

cexp 10609

cdvds 11930

cgcd 12079

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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080

This theorem is referenced by: dvdssqlem 12167 nn0gcdsq 12338 pythagtriplem3 12405

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