sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12013

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 11951	. . . . 5 $/\$
2	1	nnsqcld 10660	. . . 4 $/\$
3	2	nncnd 8922	. . 3 $/\$
4	3	mulid1d 7965	. 2 $/\$
5		nnsqcl 10575	. . . . . . 7
6	5	nnzd 9363	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10575	. . . . . . 7
9	8	nnzd 9363	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9261	. . . . . . . 8
12		nnz 9261	. . . . . . . 8
13		gcddvds 11947	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9363	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12008	. . . . . . 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 12008	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12003	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1246	. . . 4 $/\$
28		nncn 8916	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 8922	. . . . . 6 $/\$
31	1	nnap0d 8954	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10652	. . . . 5 $/\$
33		nncn 8916	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10652	. . . . 5 $/\$
36	32, 35	oveq12d 5887	. . . 4 $/\$
37		gcddiv 12003	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1241	. . . . . 6 $/\$
39	30, 31	dividapd 8732	. . . . . 6 $/\$
40	38, 39	eqtr3d 2212	. . . . 5 $/\$
41	1	nnne0d 8953	. . . . . . . . 9 $/\$
42		dvdsval2 11781	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1238	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 8915	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 8921	. . . . . . . 8 $/\$
48		nngt0 8933	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 8952	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8881	. . . . . . 7 $/\$
52		elnnz 9252	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 11781	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1238	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 8915	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 8933	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8881	. . . . . . 7 $/\$
62		elnnz 9252	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9069	. . . . . . 7
65		rppwr 12012	. . . . . . 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 8953	. . . . . . . . 9
72	71	neneqd 2368	. . . . . . . 8
73	72	intnanrd 932	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 11946	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 8922	. . . 4 $/\$
78	2	nnap0d 8954	. . . 4 $/\$ #
79		ax-1cn 7895	. . . . 5
80		divmulap 8621	. . . . 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 4000 (class class class)co 5869

cc 7800

cr 7801

cc0 7802

c1 7803

cmul 7807

clt 7982 # cap 8528

cdiv 8618

cn 8908

c2 8959

cz 9242

cexp 10505

cdvds 11778

cgcd 11926

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 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-sup 6977 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-fz 9996 df-fzo 10129 df-fl 10256 df-mod 10309 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-dvds 11779 df-gcd 11927

This theorem is referenced by: dvdssqlem 12014 nn0gcdsq 12183 pythagtriplem3 12250

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