sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 10900

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 10841	. . . . 5 $/\$
2	1	nnsqcld 10004	. . . 4 $/\$
3	2	nncnd 8371	. . 3 $/\$
4	3	mulid1d 7449	. 2 $/\$
5		nnsqcl 9923	. . . . . . 7
6	5	nnzd 8800	. . . . . 6
7	6	adantr 270	. . . . 5 $/\$
8		nnsqcl 9923	. . . . . . 7
9	8	nnzd 8800	. . . . . 6
10	9	adantl 271	. . . . 5 $/\$
11		nnz 8702	. . . . . . . 8
12		nnz 8702	. . . . . . . 8
13		gcddvds 10837	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 283	. . . . . . 7 $/\$ $/\$
15	14	simpld 110	. . . . . 6 $/\$
16	1	nnzd 8800	. . . . . . 7 $/\$
17	11	adantr 270	. . . . . . 7 $/\$
18		dvdssqim 10895	. . . . . . 7 $/\$
19	16, 17, 18	syl2anc 403	. . . . . 6 $/\$
20	15, 19	mpd 13	. . . . 5 $/\$
21	14	simprd 112	. . . . . 6 $/\$
22	12	adantl 271	. . . . . . 7 $/\$
23		dvdssqim 10895	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 403	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 10890	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1180	. . . 4 $/\$
28		nncn 8365	. . . . . . 7
29	28	adantr 270	. . . . . 6 $/\$
30	1	nncnd 8371	. . . . . 6 $/\$
31	1	nnap0d 8402	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 9996	. . . . 5 $/\$
33		nncn 8365	. . . . . . 7
34	33	adantl 271	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 9996	. . . . 5 $/\$
36	32, 35	oveq12d 5631	. . . 4 $/\$
37		gcddiv 10890	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1175	. . . . . 6 $/\$
39	30, 31	dividapd 8192	. . . . . 6 $/\$
40	38, 39	eqtr3d 2119	. . . . 5 $/\$
41	1	nnne0d 8401	. . . . . . . . 9 $/\$
42		dvdsval2 10681	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1172	. . . . . . . 8 $/\$
44	15, 43	mpbid 145	. . . . . . 7 $/\$
45		nnre 8364	. . . . . . . . 9
46	45	adantr 270	. . . . . . . 8 $/\$
47	1	nnred 8370	. . . . . . . 8 $/\$
48		nngt0 8382	. . . . . . . . 9
49	48	adantr 270	. . . . . . . 8 $/\$
50	1	nngt0d 8400	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8331	. . . . . . 7 $/\$
52		elnnz 8693	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 408	. . . . . 6 $/\$
54		dvdsval2 10681	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1172	. . . . . . . 8 $/\$
56	21, 55	mpbid 145	. . . . . . 7 $/\$
57		nnre 8364	. . . . . . . . 9
58	57	adantl 271	. . . . . . . 8 $/\$
59		nngt0 8382	. . . . . . . . 9
60	59	adantl 271	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8331	. . . . . . 7 $/\$
62		elnnz 8693	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 408	. . . . . 6 $/\$
64		2nn 8511	. . . . . . 7
65		rppwr 10899	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1260	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 403	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2123	. . 3 $/\$
70	6, 9	anim12i 331	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8401	. . . . . . . . 9
72	71	neneqd 2272	. . . . . . . 8
73	72	intnanrd 877	. . . . . . 7 $/\$
74	73	adantr 270	. . . . . 6 $/\$ $/\$
75		gcdn0cl 10836	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 403	. . . . 5 $/\$
77	76	nncnd 8371	. . . 4 $/\$
78	2	nnap0d 8402	. . . 4 $/\$ #
79		ax-1cn 7382	. . . . 5
80		divmulap 8081	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1259	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1170	. . 3 $/\$
83	69, 82	mpbid 145	. 2 $/\$
84	4, 83	eqtr3d 2119	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103

wceq 1287

wcel 1436

wne 2251 class class class wbr 3820 (class class class)co 5613

cc 7292

cr 7293

cc0 7294

c1 7295

cmul 7299

clt 7466 # cap 7999

cdiv 8078

cn 8357

c2 8407

cz 8683

cexp 9853

cdvds 10678

cgcd 10820

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3929 ax-sep 3932 ax-nul 3940 ax-pow 3984 ax-pr 4010 ax-un 4234 ax-setind 4326 ax-iinf 4376 ax-cnex 7380 ax-resscn 7381 ax-1cn 7382 ax-1re 7383 ax-icn 7384 ax-addcl 7385 ax-addrcl 7386 ax-mulcl 7387 ax-mulrcl 7388 ax-addcom 7389 ax-mulcom 7390 ax-addass 7391 ax-mulass 7392 ax-distr 7393 ax-i2m1 7394 ax-0lt1 7395 ax-1rid 7396 ax-0id 7397 ax-rnegex 7398 ax-precex 7399 ax-cnre 7400 ax-pre-ltirr 7401 ax-pre-ltwlin 7402 ax-pre-lttrn 7403 ax-pre-apti 7404 ax-pre-ltadd 7405 ax-pre-mulgt0 7406 ax-pre-mulext 7407 ax-arch 7408 ax-caucvg 7409

This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2617 df-sbc 2830 df-csb 2923 df-dif 2990 df-un 2992 df-in 2994 df-ss 3001 df-nul 3276 df-if 3380 df-pw 3417 df-sn 3437 df-pr 3438 df-op 3440 df-uni 3637 df-int 3672 df-iun 3715 df-br 3821 df-opab 3875 df-mpt 3876 df-tr 3912 df-id 4094 df-po 4097 df-iso 4098 df-iord 4167 df-on 4169 df-ilim 4170 df-suc 4172 df-iom 4379 df-xp 4417 df-rel 4418 df-cnv 4419 df-co 4420 df-dm 4421 df-rn 4422 df-res 4423 df-ima 4424 df-iota 4946 df-fun 4983 df-fn 4984 df-f 4985 df-f1 4986 df-fo 4987 df-f1o 4988 df-fv 4989 df-riota 5569 df-ov 5616 df-oprab 5617 df-mpt2 5618 df-1st 5868 df-2nd 5869 df-recs 6024 df-frec 6110 df-sup 6623 df-pnf 7468 df-mnf 7469 df-xr 7470 df-ltxr 7471 df-le 7472 df-sub 7599 df-neg 7600 df-reap 7993 df-ap 8000 df-div 8079 df-inn 8358 df-2 8416 df-3 8417 df-4 8418 df-n0 8607 df-z 8684 df-uz 8952 df-q 9037 df-rp 9067 df-fz 9357 df-fzo 9482 df-fl 9605 df-mod 9658 df-iseq 9780 df-iexp 9854 df-cj 10172 df-re 10173 df-im 10174 df-rsqrt 10327 df-abs 10328 df-dvds 10679 df-gcd 10821

This theorem is referenced by: dvdssqlem 10901 nn0gcdsq 11060

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