sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 12618

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 12556	. . . . 5 $/\$
2	1	nnsqcld 10957	. . . 4 $/\$
3	2	nncnd 9157	. . 3 $/\$
4	3	mulridd 8196	. 2 $/\$
5		nnsqcl 10872	. . . . . . 7
6	5	nnzd 9601	. . . . . 6
7	6	adantr 276	. . . . 5 $/\$
8		nnsqcl 10872	. . . . . . 7
9	8	nnzd 9601	. . . . . 6
10	9	adantl 277	. . . . 5 $/\$
11		nnz 9498	. . . . . . . 8
12		nnz 9498	. . . . . . . 8
13		gcddvds 12552	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 289	. . . . . . 7 $/\$ $/\$
15	14	simpld 112	. . . . . 6 $/\$
16	1	nnzd 9601	. . . . . . 7 $/\$
17	11	adantr 276	. . . . . . 7 $/\$
18		dvdssqim 12613	. . . . . . 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 12613	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 411	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 12608	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1281	. . . 4 $/\$
28		nncn 9151	. . . . . . 7
29	28	adantr 276	. . . . . 6 $/\$
30	1	nncnd 9157	. . . . . 6 $/\$
31	1	nnap0d 9189	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 10949	. . . . 5 $/\$
33		nncn 9151	. . . . . . 7
34	33	adantl 277	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 10949	. . . . 5 $/\$
36	32, 35	oveq12d 6036	. . . 4 $/\$
37		gcddiv 12608	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1276	. . . . . 6 $/\$
39	30, 31	dividapd 8966	. . . . . 6 $/\$
40	38, 39	eqtr3d 2266	. . . . 5 $/\$
41	1	nnne0d 9188	. . . . . . . . 9 $/\$
42		dvdsval2 12369	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1273	. . . . . . . 8 $/\$
44	15, 43	mpbid 147	. . . . . . 7 $/\$
45		nnre 9150	. . . . . . . . 9
46	45	adantr 276	. . . . . . . 8 $/\$
47	1	nnred 9156	. . . . . . . 8 $/\$
48		nngt0 9168	. . . . . . . . 9
49	48	adantr 276	. . . . . . . 8 $/\$
50	1	nngt0d 9187	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 9115	. . . . . . 7 $/\$
52		elnnz 9489	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 417	. . . . . 6 $/\$
54		dvdsval2 12369	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1273	. . . . . . . 8 $/\$
56	21, 55	mpbid 147	. . . . . . 7 $/\$
57		nnre 9150	. . . . . . . . 9
58	57	adantl 277	. . . . . . . 8 $/\$
59		nngt0 9168	. . . . . . . . 9
60	59	adantl 277	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 9115	. . . . . . 7 $/\$
62		elnnz 9489	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 417	. . . . . 6 $/\$
64		2nn 9305	. . . . . . 7
65		rppwr 12617	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1362	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 411	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2270	. . 3 $/\$
70	6, 9	anim12i 338	. . . . . 6 $/\$ $/\$
71	5	nnne0d 9188	. . . . . . . . 9
72	71	neneqd 2423	. . . . . . . 8
73	72	intnanrd 939	. . . . . . 7 $/\$
74	73	adantr 276	. . . . . 6 $/\$ $/\$
75		gcdn0cl 12551	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 411	. . . . 5 $/\$
77	76	nncnd 9157	. . . 4 $/\$
78	2	nnap0d 9189	. . . 4 $/\$ #
79		ax-1cn 8125	. . . . 5
80		divmulap 8855	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1361	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1271	. . 3 $/\$
83	69, 82	mpbid 147	. 2 $/\$
84	4, 83	eqtr3d 2266	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1397

wcel 2202

wne 2402 class class class wbr 4088 (class class class)co 6018

cc 8030

cr 8031

cc0 8032

c1 8033

cmul 8037

clt 8214 # cap 8761

cdiv 8852

cn 9143

c2 9194

cz 9479

cexp 10801

cdvds 12366

cgcd 12542

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-sup 7183 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fz 10244 df-fzo 10378 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-cj 11420 df-re 11421 df-im 11422 df-rsqrt 11576 df-abs 11577 df-dvds 12367 df-gcd 12543

This theorem is referenced by: dvdssqlem 12619 nn0gcdsq 12790 pythagtriplem3 12858

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