sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12350

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 12288	. . . . 5 $/\$
2	1	nnsqcld 10839	. . . 4 $/\$
3	2	nncnd 9050	. . 3 $/\$
4	3	mulridd 8089	. 2 $/\$
5		nnsqcl 10754	. . . . . . 7
6	5	nnzd 9494	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10754	. . . . . . 7
9	8	nnzd 9494	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9391	. . . . . . . 8
12		nnz 9391	. . . . . . . 8
13		gcddvds 12284	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9494	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12345	. . . . . . 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 12345	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12340	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1258	. . . 4 $/\$
28		nncn 9044	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 9050	. . . . . 6 $/\$
31	1	nnap0d 9082	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10831	. . . . 5 $/\$
33		nncn 9044	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10831	. . . . 5 $/\$
36	32, 35	oveq12d 5962	. . . 4 $/\$
37		gcddiv 12340	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1253	. . . . . 6 $/\$
39	30, 31	dividapd 8859	. . . . . 6 $/\$
40	38, 39	eqtr3d 2240	. . . . 5 $/\$
41	1	nnne0d 9081	. . . . . . . . 9 $/\$
42		dvdsval2 12101	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1250	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 9043	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 9049	. . . . . . . 8 $/\$
48		nngt0 9061	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 9080	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 9008	. . . . . . 7 $/\$
52		elnnz 9382	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 12101	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1250	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 9043	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 9061	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 9008	. . . . . . 7 $/\$
62		elnnz 9382	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9198	. . . . . . 7
65		rppwr 12349	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1339	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 411	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2244	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 9081	. . . . . . . . 9
72	71	neneqd 2397	. . . . . . . 8
73	72	intnanrd 934	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 12283	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 9050	. . . 4 $/\$
78	2	nnap0d 9082	. . . 4 $/\$ #
79		ax-1cn 8018	. . . . 5
80		divmulap 8748	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1338	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1248	. . 3 $/\$
83	69, 82	mpbid 147	. 2 $/\$
84	4, 83	eqtr3d 2240	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1373

wcel 2176

wne 2376 class class class wbr 4044 (class class class)co 5944

cc 7923

cr 7924

cc0 7925

c1 7926

cmul 7930

clt 8107 # cap 8654

cdiv 8745

cn 9036

c2 9087

cz 9372

cexp 10683

cdvds 12098

cgcd 12274

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-sup 7086 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-fz 10131 df-fzo 10265 df-fl 10413 df-mod 10468 df-seqfrec 10593 df-exp 10684 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-dvds 12099 df-gcd 12275

This theorem is referenced by: dvdssqlem 12351 nn0gcdsq 12522 pythagtriplem3 12590

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