sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 11947

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 11885	. . . . 5 $/\$
2	1	nnsqcld 10598	. . . 4 $/\$
3	2	nncnd 8862	. . 3 $/\$
4	3	mulid1d 7907	. 2 $/\$
5		nnsqcl 10514	. . . . . . 7
6	5	nnzd 9303	. . . . . 6
7	6	adantr 274	. . . . 5 $/\$
8		nnsqcl 10514	. . . . . . 7
9	8	nnzd 9303	. . . . . 6
10	9	adantl 275	. . . . 5 $/\$
11		nnz 9201	. . . . . . . 8
12		nnz 9201	. . . . . . . 8
13		gcddvds 11881	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 287	. . . . . . 7 $/\$ $/\$
15	14	simpld 111	. . . . . 6 $/\$
16	1	nnzd 9303	. . . . . . 7 $/\$
17	11	adantr 274	. . . . . . 7 $/\$
18		dvdssqim 11942	. . . . . . 7 $/\$
19	16, 17, 18	syl2anc 409	. . . . . 6 $/\$
20	15, 19	mpd 13	. . . . 5 $/\$
21	14	simprd 113	. . . . . 6 $/\$
22	12	adantl 275	. . . . . . 7 $/\$
23		dvdssqim 11942	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 409	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 11937	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1235	. . . 4 $/\$
28		nncn 8856	. . . . . . 7
29	28	adantr 274	. . . . . 6 $/\$
30	1	nncnd 8862	. . . . . 6 $/\$
31	1	nnap0d 8894	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10590	. . . . 5 $/\$
33		nncn 8856	. . . . . . 7
34	33	adantl 275	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10590	. . . . 5 $/\$
36	32, 35	oveq12d 5854	. . . 4 $/\$
37		gcddiv 11937	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1230	. . . . . 6 $/\$
39	30, 31	dividapd 8673	. . . . . 6 $/\$
40	38, 39	eqtr3d 2199	. . . . 5 $/\$
41	1	nnne0d 8893	. . . . . . . . 9 $/\$
42		dvdsval2 11716	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1227	. . . . . . . 8 $/\$
44	15, 43	mpbid 146	. . . . . . 7 $/\$
45		nnre 8855	. . . . . . . . 9
46	45	adantr 274	. . . . . . . 8 $/\$
47	1	nnred 8861	. . . . . . . 8 $/\$
48		nngt0 8873	. . . . . . . . 9
49	48	adantr 274	. . . . . . . 8 $/\$
50	1	nngt0d 8892	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8821	. . . . . . 7 $/\$
52		elnnz 9192	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 414	. . . . . 6 $/\$
54		dvdsval2 11716	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1227	. . . . . . . 8 $/\$
56	21, 55	mpbid 146	. . . . . . 7 $/\$
57		nnre 8855	. . . . . . . . 9
58	57	adantl 275	. . . . . . . 8 $/\$
59		nngt0 8873	. . . . . . . . 9
60	59	adantl 275	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8821	. . . . . . 7 $/\$
62		elnnz 9192	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 414	. . . . . 6 $/\$
64		2nn 9009	. . . . . . 7
65		rppwr 11946	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1315	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 409	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2203	. . 3 $/\$
70	6, 9	anim12i 336	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8893	. . . . . . . . 9
72	71	neneqd 2355	. . . . . . . 8
73	72	intnanrd 922	. . . . . . 7 $/\$
74	73	adantr 274	. . . . . 6 $/\$ $/\$
75		gcdn0cl 11880	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 409	. . . . 5 $/\$
77	76	nncnd 8862	. . . 4 $/\$
78	2	nnap0d 8894	. . . 4 $/\$ #
79		ax-1cn 7837	. . . . 5
80		divmulap 8562	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1314	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1225	. . 3 $/\$
83	69, 82	mpbid 146	. 2 $/\$
84	4, 83	eqtr3d 2199	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104

wceq 1342

wcel 2135

wne 2334 class class class wbr 3976 (class class class)co 5836

cc 7742

cr 7743

cc0 7744

c1 7745

cmul 7749

clt 7924 # cap 8470

cdiv 8559

cn 8848

c2 8899

cz 9182

cexp 10444

cdvds 11713

cgcd 11860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861

This theorem is referenced by: dvdssqlem 11948 nn0gcdsq 12109 pythagtriplem3 12176

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