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Theorem prmreclem1 14434

Description: Lemma for prmrec 14440. Properties of the "square part" function, which extracts the of the decomposition N=2, with maximal and squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)

**Hypothesis**
Ref	Expression
prmreclem1.1

**Assertion**
Ref	Expression
prmreclem1

Distinct variable groups: , ,,N Q,

Proof of Theorem prmreclem1 Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3584	. . 3
2		breq2 4456	. . . . . . 7
3	2	rabbidv 3101	. . . . . 6
4	3	supeq1d 7926	. . . . 5
5		prmreclem1.1	. . . . 5
6		ltso 9686	. . . . . 6
7	6	supex 7943	. . . . 5
8	4, 5, 7	fvmpt 5956	. . . 4
9		nnssz 10909	. . . . . . 7
10	1, 9	sstri 3512	. . . . . 6
11	10	a1i 11	. . . . 5
12		1nn 10572	. . . . . . . 8
13	12	a1i 11	. . . . . . 7
14		nnz 10911	. . . . . . . 8
15		1dvds 13998	. . . . . . . 8
16	14, 15	syl 16	. . . . . . 7
17		oveq1 6303	. . . . . . . . . 10
18		sq1 12262	. . . . . . . . . 10
19	17, 18	syl6eq 2514	. . . . . . . . 9
20	19	breq1d 4462	. . . . . . . 8
21	20	elrab 3257	. . . . . . 7
22	13, 16, 21	sylanbrc 664	. . . . . 6
23		ne0i 3790	. . . . . 6
24	22, 23	syl 16	. . . . 5
25		nnz 10911	. . . . . . . . . . 11
26		zsqcl 12238	. . . . . . . . . . 11
27	25, 26	syl 16	. . . . . . . . . 10
28		id 22	. . . . . . . . . 10
29		dvdsle 14031	. . . . . . . . . 10
30	27, 28, 29	syl2anr 478	. . . . . . . . 9
31		nnlesq 12271	. . . . . . . . . . 11
32	31	adantl 466	. . . . . . . . . 10
33		nnre 10568	. . . . . . . . . . . 12
34	33	adantl 466	. . . . . . . . . . 11
35	34	resqcld 12336	. . . . . . . . . . 11
36		nnre 10568	. . . . . . . . . . . 12
37	36	adantr 465	. . . . . . . . . . 11
38		letr 9699	. . . . . . . . . . 11
39	34, 35, 37, 38	syl3anc 1228	. . . . . . . . . 10
40	32, 39	mpand 675	. . . . . . . . 9
41	30, 40	syld 44	. . . . . . . 8
42	41	ralrimiva 2871	. . . . . . 7
43		oveq1 6303	. . . . . . . . 9
44	43	breq1d 4462	. . . . . . . 8
45	44	ralrab 3261	. . . . . . 7
46	42, 45	sylibr 212	. . . . . 6
47		breq2 4456	. . . . . . . 8
48	47	ralbidv 2896	. . . . . . 7
49	48	rspcev 3210	. . . . . 6
50	14, 46, 49	syl2anc 661	. . . . 5
51		suprzcl2 11201	. . . . 5
52	11, 24, 50, 51	syl3anc 1228	. . . 4
53	8, 52	eqeltrd 2545	. . 3
54	1, 53	sseldi 3501	. 2
55		oveq1 6303	. . . . . 6
56	55	breq1d 4462	. . . . 5
57	44	cbvrabv 3108	. . . . 5
58	56, 57	elrab2 3259	. . . 4
59	53, 58	sylib 196	. . 3
60	59	simprd 463	. 2
61	54	adantr 465	. . . . . . . 8
62	61	nncnd 10577	. . . . . . 7
63	62	mulid1d 9634	. . . . . 6
64		eluz2b2 11183	. . . . . . . . 9
65	64	simprbi 464	. . . . . . . 8
66	65	adantl 466	. . . . . . 7
67		1red 9632	. . . . . . . 8
68		eluz2nn 11148	. . . . . . . . . 10
69	68	adantl 466	. . . . . . . . 9
70	69	nnred 10576	. . . . . . . 8
71	61	nnred 10576	. . . . . . . 8
72	61	nngt0d 10604	. . . . . . . 8
73		ltmul2 10418	. . . . . . . 8
74	67, 70, 71, 72, 73	syl112anc 1232	. . . . . . 7
75	66, 74	mpbid 210	. . . . . 6
76	63, 75	eqbrtrrd 4474	. . . . 5
77		nnmulcl 10584	. . . . . . . 8
78	54, 68, 77	syl2an 477	. . . . . . 7
79	78	nnred 10576	. . . . . 6
80	71, 79	ltnled 9753	. . . . 5
81	76, 80	mpbid 210	. . . 4
82	10	a1i 11	. . . . . 6
83	50	ad2antrr 725	. . . . . 6
84	78	adantr 465	. . . . . . 7
85		simpr 461	. . . . . . . . 9
86	69	adantr 465	. . . . . . . . . . . 12
87	86	nnsqcld 12330	. . . . . . . . . . 11
88		nnz 10911	. . . . . . . . . . 11
89	87, 88	syl 16	. . . . . . . . . 10
90	54	nnsqcld 12330	. . . . . . . . . . . . . 14
91	9, 90	sseldi 3501	. . . . . . . . . . . . 13
92	90	nnne0d 10605	. . . . . . . . . . . . 13
93		dvdsval2 13989	. . . . . . . . . . . . 13
94	91, 92, 14, 93	syl3anc 1228	. . . . . . . . . . . 12
95	60, 94	mpbid 210	. . . . . . . . . . 11
96	95	ad2antrr 725	. . . . . . . . . 10
97	91	ad2antrr 725	. . . . . . . . . 10
98		dvdscmul 14010	. . . . . . . . . 10
99	89, 96, 97, 98	syl3anc 1228	. . . . . . . . 9
100	85, 99	mpd 15	. . . . . . . 8
101	62	adantr 465	. . . . . . . . . 10
102	86	nncnd 10577	. . . . . . . . . 10
103	101, 102	sqmuld 12322	. . . . . . . . 9
104	103	eqcomd 2465	. . . . . . . 8
105		nncn 10569	. . . . . . . . . 10
106	105	ad2antrr 725	. . . . . . . . 9
107	90	ad2antrr 725	. . . . . . . . . 10
108	107	nncnd 10577	. . . . . . . . 9
109	92	ad2antrr 725	. . . . . . . . 9
110	106, 108, 109	divcan2d 10347	. . . . . . . 8
111	100, 104, 110	3brtr3d 4481	. . . . . . 7
112		oveq1 6303	. . . . . . . . 9
113	112	breq1d 4462	. . . . . . . 8
114	113	elrab 3257	. . . . . . 7
115	84, 111, 114	sylanbrc 664	. . . . . 6
116		suprzub 11202	. . . . . 6
117	82, 83, 115, 116	syl3anc 1228	. . . . 5
118	8	ad2antrr 725	. . . . 5
119	117, 118	breqtrrd 4478	. . . 4
120	81, 119	mtand 659	. . 3
121	120	ex 434	. 2
122	54, 60, 121	3jca 1176	1

Colors of variables: wff setvar class

Syntax hints: -.wn 3 ->wi 4 <->wb 184 /\wa 369 /\w3a 973 =wceq 1395 e.wcel 1818 =/=wne 2652 A.wral 2807 E.wrex 2808 {crab 2811 C_wss 3475 c0 3784 class class class wbr 4452 e.cmpt 4510 `cfv 5593 (class class class)co 6296 supcsup 7920 cc 9511 cr 9512 0cc0 9513 1c1 9514 cmul 9518 clt 9649 cle 9650 cdiv 10231 cn 10561 2c2 10610 cz 10889 cuz 11110 cexp 12166 cdvds 13986

This theorem is referenced by: prmreclem2 14435 prmreclem3 14436

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-2nd 6801 df-recs 7061 df-rdg 7095 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-sup 7921 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-2 10619 df-n0 10821 df-z 10890 df-uz 11111 df-seq 12108 df-exp 12167 df-dvds 13987

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