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Theorem hashxplem 13220

Description: Lemma for hashxp 13221. (Contributed by Paul Chapman, 30-Nov-2012.)

**Hypothesis**
Ref	Expression
hashxplem.1

**Assertion**
Ref	Expression
hashxplem

Proof of Theorem hashxplem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 5128	. . . 4
2	1	fveq2d 6195	. . 3
3		fveq2 6191	. . . 4
4	3	oveq1d 6665	. . 3
5	2, 4	eqeq12d 2637	. 2
6		xpeq1 5128	. . . 4
7	6	fveq2d 6195	. . 3
8		fveq2 6191	. . . 4
9	8	oveq1d 6665	. . 3
10	7, 9	eqeq12d 2637	. 2
11		xpeq1 5128	. . . 4
12	11	fveq2d 6195	. . 3
13		fveq2 6191	. . . 4
14	13	oveq1d 6665	. . 3
15	12, 14	eqeq12d 2637	. 2
16		xpeq1 5128	. . . 4
17	16	fveq2d 6195	. . 3
18		fveq2 6191	. . . 4
19	18	oveq1d 6665	. . 3
20	17, 19	eqeq12d 2637	. 2
21		hashxplem.1	. . . 4
22		hashcl 13147	. . . . . 6
23	22	nn0cnd 11353	. . . . 5
24	23	mul02d 10234	. . . 4
25	21, 24	ax-mp 5	. . 3
26		hash0 13158	. . . 4
27	26	oveq1i 6660	. . 3
28		0xp 5199	. . . . 5
29	28	fveq2i 6194	. . . 4
30	29, 26	eqtri 2644	. . 3
31	25, 27, 30	3eqtr4ri 2655	. 2
32		oveq1 6657	. . . . 5
33	32	adantl 482	. . . 4 $/\$ $/\$
34		xpundir 5172	. . . . . . 7
35	34	fveq2i 6194	. . . . . 6
36		xpfi 8231	. . . . . . . . 9 $/\$
37	21, 36	mpan2 707	. . . . . . . 8
38		inxp 5254	. . . . . . . . 9
39		disjsn 4246	. . . . . . . . . . . 12
40	39	biimpri 218	. . . . . . . . . . 11
41	40	xpeq1d 5138	. . . . . . . . . 10
42		0xp 5199	. . . . . . . . . 10
43	41, 42	syl6eq 2672	. . . . . . . . 9
44	38, 43	syl5eq 2668	. . . . . . . 8
45		snfi 8038	. . . . . . . . . 10
46		xpfi 8231	. . . . . . . . . 10 $/\$
47	45, 21, 46	mp2an 708	. . . . . . . . 9
48		hashun 13171	. . . . . . . . 9 $/\$ $/\$
49	47, 48	mp3an2 1412	. . . . . . . 8 $/\$
50	37, 44, 49	syl2an 494	. . . . . . 7 $/\$
51		snex 4908	. . . . . . . . . . 11
52	21	elexi 3213	. . . . . . . . . . 11
53	51, 52	xpcomen 8051	. . . . . . . . . 10
54		vex 3203	. . . . . . . . . . 11
55	52, 54	xpsnen 8044	. . . . . . . . . 10
56	53, 55	entri 8010	. . . . . . . . 9
57		hashen 13135	. . . . . . . . . 10 $/\$
58	47, 21, 57	mp2an 708	. . . . . . . . 9
59	56, 58	mpbir 221	. . . . . . . 8
60	59	oveq2i 6661	. . . . . . 7
61	50, 60	syl6eq 2672	. . . . . 6 $/\$
62	35, 61	syl5eq 2668	. . . . 5 $/\$
63	62	adantr 481	. . . 4 $/\$ $/\$
64		hashunsng 13181	. . . . . . . 8 $/\$
65	54, 64	ax-mp 5	. . . . . . 7 $/\$
66	65	oveq1d 6665	. . . . . 6 $/\$
67		hashcl 13147	. . . . . . . . . 10
68	67	nn0cnd 11353	. . . . . . . . 9
69		ax-1cn 9994	. . . . . . . . . 10
70		nn0cn 11302	. . . . . . . . . . 11
71	21, 22, 70	mp2b 10	. . . . . . . . . 10
72		adddir 10031	. . . . . . . . . 10 $/\$ $/\$
73	69, 71, 72	mp3an23 1416	. . . . . . . . 9
74	68, 73	syl 17	. . . . . . . 8
75	71	mulid2i 10043	. . . . . . . . 9
76	75	oveq2i 6661	. . . . . . . 8
77	74, 76	syl6eq 2672	. . . . . . 7
78	77	adantr 481	. . . . . 6 $/\$
79	66, 78	eqtrd 2656	. . . . 5 $/\$
80	79	adantr 481	. . . 4 $/\$ $/\$
81	33, 63, 80	3eqtr4d 2666	. . 3 $/\$ $/\$
82	81	ex 450	. 2 $/\$
83	5, 10, 15, 20, 31, 82	findcard2s 8201	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

cvv 3200

cun 3572

cin 3573

c0 3915

csn 4177 class class class wbr 4653

cxp 5112

cfv 5888 (class class class)co 6650

cen 7952

cfn 7955

cc 9934

cc0 9936

c1 9937

caddc 9939

cmul 9941

cn0 11292

chash 13117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118

This theorem is referenced by: hashxp 13221

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