sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12032

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 11970	. . . . 5 $/\$
2	1	nnsqcld 10677	. . . 4 $/\$
3	2	nncnd 8935	. . 3 $/\$
4	3	mulridd 7976	. 2 $/\$
5		nnsqcl 10592	. . . . . . 7
6	5	nnzd 9376	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10592	. . . . . . 7
9	8	nnzd 9376	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9274	. . . . . . . 8
12		nnz 9274	. . . . . . . 8
13		gcddvds 11966	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9376	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12027	. . . . . . 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 12027	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12022	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1246	. . . 4 $/\$
28		nncn 8929	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 8935	. . . . . 6 $/\$
31	1	nnap0d 8967	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10669	. . . . 5 $/\$
33		nncn 8929	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10669	. . . . 5 $/\$
36	32, 35	oveq12d 5895	. . . 4 $/\$
37		gcddiv 12022	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1241	. . . . . 6 $/\$
39	30, 31	dividapd 8745	. . . . . 6 $/\$
40	38, 39	eqtr3d 2212	. . . . 5 $/\$
41	1	nnne0d 8966	. . . . . . . . 9 $/\$
42		dvdsval2 11799	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1238	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 8928	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 8934	. . . . . . . 8 $/\$
48		nngt0 8946	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 8965	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8894	. . . . . . 7 $/\$
52		elnnz 9265	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 11799	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1238	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 8928	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 8946	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8894	. . . . . . 7 $/\$
62		elnnz 9265	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9082	. . . . . . 7
65		rppwr 12031	. . . . . . 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 2216	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8966	. . . . . . . . 9
72	71	neneqd 2368	. . . . . . . 8
73	72	intnanrd 932	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 11965	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 8935	. . . 4 $/\$
78	2	nnap0d 8967	. . . 4 $/\$ #
79		ax-1cn 7906	. . . . 5
80		divmulap 8634	. . . . 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 2212	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

wne 2347 class class class wbr 4005 (class class class)co 5877

cc 7811

cr 7812

cc0 7813

c1 7814

cmul 7818

clt 7994 # cap 8540

cdiv 8631

cn 8921

c2 8972

cz 9255

cexp 10521

cdvds 11796

cgcd 11945

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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

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 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-sup 6985 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-fz 10011 df-fzo 10145 df-fl 10272 df-mod 10325 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-dvds 11797 df-gcd 11946

This theorem is referenced by: dvdssqlem 12033 nn0gcdsq 12202 pythagtriplem3 12269

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