sqgcd - Intuitionistic Logic Explorer

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Theorem sqgcd 10418

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 10359	. . . . 5 $/\$
2	1	nnsqcld 9626	. . . 4 $/\$
3	2	nncnd 8053	. . 3 $/\$
4	3	mulid1d 7136	. 2 $/\$
5		nnsqcl 9545	. . . . . . 7
6	5	nnzd 8468	. . . . . 6
7	6	adantr 270	. . . . 5 $/\$
8		nnsqcl 9545	. . . . . . 7
9	8	nnzd 8468	. . . . . 6
10	9	adantl 271	. . . . 5 $/\$
11		nnz 8370	. . . . . . . 8
12		nnz 8370	. . . . . . . 8
13		gcddvds 10355	. . . . . . . 8 $/\$ $/\$
14	11, 12, 13	syl2an 283	. . . . . . 7 $/\$ $/\$
15	14	simpld 110	. . . . . 6 $/\$
16	1	nnzd 8468	. . . . . . 7 $/\$
17	11	adantr 270	. . . . . . 7 $/\$
18		dvdssqim 10413	. . . . . . 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 10413	. . . . . . 7 $/\$
24	16, 22, 23	syl2anc 403	. . . . . 6 $/\$
25	21, 24	mpd 13	. . . . 5 $/\$
26		gcddiv 10408	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	7, 10, 2, 20, 25, 26	syl32anc 1177	. . . 4 $/\$
28		nncn 8047	. . . . . . 7
29	28	adantr 270	. . . . . 6 $/\$
30	1	nncnd 8053	. . . . . 6 $/\$
31	1	nnap0d 8084	. . . . . 6 $/\$ #
32	29, 30, 31	sqdivapd 9618	. . . . 5 $/\$
33		nncn 8047	. . . . . . 7
34	33	adantl 271	. . . . . 6 $/\$
35	34, 30, 31	sqdivapd 9618	. . . . 5 $/\$
36	32, 35	oveq12d 5550	. . . 4 $/\$
37		gcddiv 10408	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	17, 22, 1, 14, 37	syl31anc 1172	. . . . . 6 $/\$
39	30, 31	dividapd 7874	. . . . . 6 $/\$
40	38, 39	eqtr3d 2115	. . . . 5 $/\$
41	1	nnne0d 8083	. . . . . . . . 9 $/\$
42		dvdsval2 10198	. . . . . . . . 9 $/\$ $/\$
43	16, 41, 17, 42	syl3anc 1169	. . . . . . . 8 $/\$
44	15, 43	mpbid 145	. . . . . . 7 $/\$
45		nnre 8046	. . . . . . . . 9
46	45	adantr 270	. . . . . . . 8 $/\$
47	1	nnred 8052	. . . . . . . 8 $/\$
48		nngt0 8064	. . . . . . . . 9
49	48	adantr 270	. . . . . . . 8 $/\$
50	1	nngt0d 8082	. . . . . . . 8 $/\$
51	46, 47, 49, 50	divgt0d 8013	. . . . . . 7 $/\$
52		elnnz 8361	. . . . . . 7 $/\$
53	44, 51, 52	sylanbrc 408	. . . . . 6 $/\$
54		dvdsval2 10198	. . . . . . . . 9 $/\$ $/\$
55	16, 41, 22, 54	syl3anc 1169	. . . . . . . 8 $/\$
56	21, 55	mpbid 145	. . . . . . 7 $/\$
57		nnre 8046	. . . . . . . . 9
58	57	adantl 271	. . . . . . . 8 $/\$
59		nngt0 8064	. . . . . . . . 9
60	59	adantl 271	. . . . . . . 8 $/\$
61	58, 47, 60, 50	divgt0d 8013	. . . . . . 7 $/\$
62		elnnz 8361	. . . . . . 7 $/\$
63	56, 61, 62	sylanbrc 408	. . . . . 6 $/\$
64		2nn 8193	. . . . . . 7
65		rppwr 10417	. . . . . . 7 $/\$ $/\$
66	64, 65	mp3an3 1257	. . . . . 6 $/\$
67	53, 63, 66	syl2anc 403	. . . . 5 $/\$
68	40, 67	mpd 13	. . . 4 $/\$
69	27, 36, 68	3eqtr2d 2119	. . 3 $/\$
70	6, 9	anim12i 331	. . . . . 6 $/\$ $/\$
71	5	nnne0d 8083	. . . . . . . . 9
72	71	neneqd 2266	. . . . . . . 8
73	72	intnanrd 874	. . . . . . 7 $/\$
74	73	adantr 270	. . . . . 6 $/\$ $/\$
75		gcdn0cl 10354	. . . . . 6 $/\$ $/\$ $/\$
76	70, 74, 75	syl2anc 403	. . . . 5 $/\$
77	76	nncnd 8053	. . . 4 $/\$
78	2	nnap0d 8084	. . . 4 $/\$ #
79		ax-1cn 7069	. . . . 5
80		divmulap 7763	. . . . 5 $/\$ $/\$ $/\$ #
81	79, 80	mp3an2 1256	. . . 4 $/\$ $/\$ #
82	77, 3, 78, 81	syl12anc 1167	. . 3 $/\$
83	69, 82	mpbid 145	. 2 $/\$
84	4, 83	eqtr3d 2115	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102

wb 103

wceq 1284

wcel 1433

wne 2245 class class class wbr 3785 (class class class)co 5532

cc 6979

cr 6980

cc0 6981

c1 6982

cmul 6986

clt 7153 # cap 7681

cdiv 7760

cn 8039

c2 8089

cz 8351

cexp 9475

cdvds 10195

cgcd 10338

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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-sup 6397 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-fzo 9153 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-dvds 10196 df-gcd 10339

This theorem is referenced by: dvdssqlem 10419

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