sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12831

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 12769	. . . . 5 $/\$
2	1	nnsqcld 11153	. . . 4 $/\$
3	2	nncnd 9327	. . 3 $/\$
4	3	mulridd 8349	. 2 $/\$
5		nnsqcl 11067	. . . . . . 7
6	5	nnzd 9778	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 11067	. . . . . . 7
9	8	nnzd 9778	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9674	. . . . . . . 8
12		nnz 9674	. . . . . . . 8
13		gcddvds 12765	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9778	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12826	. . . . . . 7 $/\$
19	16, 17, 18	syl2anc 415	. . . . . 6 $/\$
20	15, 19	mpd 13	. . . . 5 $/\$
21	14	simprd 114	. . . . . 6 $/\$
22	12	adantl 277	. . . . . . 7 $/\$
23		dvdssqim 12826	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 415	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12821	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1286	. . . 4 $/\$
28		nncn 9321	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 9327	. . . . . 6 $/\$
31	1	nnap0d 9359	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 11145	. . . . 5 $/\$
33		nncn 9321	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 11145	. . . . 5 $/\$
36	32, 35	oveq12d 6106	. . . 4 $/\$
37		gcddiv 12821	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1281	. . . . . 6 $/\$
39	30, 31	dividapd 9124	. . . . . 6 $/\$
40	38, 39	eqtr3d 2273	. . . . 5 $/\$
41	1	nnne0d 9358	. . . . . . . . 9 $/\$
42		dvdsval2 12582	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1278	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 9320	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 9326	. . . . . . . 8 $/\$
48		nngt0 9338	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 9357	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 9273	. . . . . . 7 $/\$
52		elnnz 9665	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 421	. . . . . 6 $/\$
54		dvdsval2 12582	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1278	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 9320	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 9338	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 9273	. . . . . . 7 $/\$
62		elnnz 9665	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 421	. . . . . 6 $/\$
64		2nn 9477	. . . . . . 7
65		rppwr 12830	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1367	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 415	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2277	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 9358	. . . . . . . . 9
72	71	neneqd 2441	. . . . . . . 8
73	72	intnanrd 944	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 12764	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 415	. . . . 5 $/\$
77	76	nncnd 9327	. . . 4 $/\$
78	2	nnap0d 9359	. . . 4 $/\$ #
79		ax-1cn 8278	. . . . 5
80		divmulap 9013	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1366	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1276	. . 3 $/\$
83	69, 82	mpbid 147	. 2 $/\$
84	4, 83	eqtr3d 2273	1 $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1402

wcel 2209

wne 2420 class class class wbr 4131 (class class class)co 6088

cc 8183

cr 8184

cc0 8185

c1 8186

cmul 8190

clt 8366 # cap 8917

cdiv 9010

cn 9313

c2 9364

cz 9655

cexp 10996

cdvds 12579

cgcd 12755

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7330 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-dvds 12580 df-gcd 12756

This theorem is used by: dvdssqlem 12832 nn0gcdsq 13005 pythagtriplem3 13075

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