sqgcd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > sqgcd		Unicode version

Theorem sqgcd 10798

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 10739	. . . . 5 $/\$
2	1	nnsqcld 9942	. . . 4 $/\$
3	2	nncnd 8330	. . 3 $/\$
4	3	mulid1d 7408	. 2 $/\$
5		nnsqcl 9861	. . . . . . 7
6	5	nnzd 8763	. . . . . 6
7	6	adantr 270	. . . . 5 $/\$
8		nnsqcl 9861	. . . . . . 7
9	8	nnzd 8763	. . . . . 6
10	9	adantl 271	. . . . 5 $/\$
11		nnz 8665	. . . . . . . 8
12		nnz 8665	. . . . . . . 8
13		gcddvds 10735	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 283	. . . . . . 7 $/\$ $/\$
15	14	simpld 110	. . . . . 6 $/\$
16	1	nnzd 8763	. . . . . . 7 $/\$
17	11	adantr 270	. . . . . . 7 $/\$
18		dvdssqim 10793	. . . . . . 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 10793	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 403	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 10788	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1178	. . . 4 $/\$
28		nncn 8324	. . . . . . 7
29	28	adantr 270	. . . . . 6 $/\$
30	1	nncnd 8330	. . . . . 6 $/\$
31	1	nnap0d 8361	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 9934	. . . . 5 $/\$
33		nncn 8324	. . . . . . 7
34	33	adantl 271	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 9934	. . . . 5 $/\$
36	32, 35	oveq12d 5609	. . . 4 $/\$
37		gcddiv 10788	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1173	. . . . . 6 $/\$
39	30, 31	dividapd 8151	. . . . . 6 $/\$
40	38, 39	eqtr3d 2117	. . . . 5 $/\$
41	1	nnne0d 8360	. . . . . . . . 9 $/\$
42		dvdsval2 10579	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1170	. . . . . . . 8 $/\$
44	15, 43	mpbid 145	. . . . . . 7 $/\$
45		nnre 8323	. . . . . . . . 9
46	45	adantr 270	. . . . . . . 8 $/\$
47	1	nnred 8329	. . . . . . . 8 $/\$
48		nngt0 8341	. . . . . . . . 9
49	48	adantr 270	. . . . . . . 8 $/\$
50	1	nngt0d 8359	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8290	. . . . . . 7 $/\$
52		elnnz 8656	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 408	. . . . . 6 $/\$
54		dvdsval2 10579	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1170	. . . . . . . 8 $/\$
56	21, 55	mpbid 145	. . . . . . 7 $/\$
57		nnre 8323	. . . . . . . . 9
58	57	adantl 271	. . . . . . . 8 $/\$
59		nngt0 8341	. . . . . . . . 9
60	59	adantl 271	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8290	. . . . . . 7 $/\$
62		elnnz 8656	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 408	. . . . . 6 $/\$
64		2nn 8470	. . . . . . 7
65		rppwr 10797	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1258	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 403	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2121	. . 3 $/\$
70	6, 9	anim12i 331	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8360	. . . . . . . . 9
72	71	neneqd 2270	. . . . . . . 8
73	72	intnanrd 875	. . . . . . 7 $/\$
74	73	adantr 270	. . . . . 6 $/\$ $/\$
75		gcdn0cl 10734	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 403	. . . . 5 $/\$
77	76	nncnd 8330	. . . 4 $/\$
78	2	nnap0d 8361	. . . 4 $/\$ #
79		ax-1cn 7341	. . . . 5
80		divmulap 8040	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1257	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1168	. . 3 $/\$
83	69, 82	mpbid 145	. 2 $/\$
84	4, 83	eqtr3d 2117	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103

wceq 1285

wcel 1434

wne 2249 class class class wbr 3811 (class class class)co 5591

cc 7251

cr 7252

cc0 7253

c1 7254

cmul 7258

clt 7425 # cap 7958

cdiv 8037

cn 8316

c2 8366

cz 8646

cexp 9791

cdvds 10576

cgcd 10718

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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 ax-arch 7367 ax-caucvg 7368

This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-sup 6586 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-2 8375 df-3 8376 df-4 8377 df-n0 8566 df-z 8647 df-uz 8915 df-q 9000 df-rp 9030 df-fz 9320 df-fzo 9444 df-fl 9566 df-mod 9619 df-iseq 9741 df-iexp 9792 df-cj 10103 df-re 10104 df-im 10105 df-rsqrt 10258 df-abs 10259 df-dvds 10577 df-gcd 10719

This theorem is referenced by: dvdssqlem 10799 nn0gcdsq 10958

W3C validator