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Theorem hashdom 13168

Description: Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

**Assertion**
Ref	Expression
hashdom	$/\$

Proof of Theorem hashdom Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 12771	. . . . . . . 8
2		ficardom 8787	. . . . . . . 8
3	1, 2	ax-mp 5	. . . . . . 7
4		eqid 2622	. . . . . . . . . . . . . 14
5	4	hashgval 13120	. . . . . . . . . . . . 13
6	5	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$
7	4	hashgval 13120	. . . . . . . . . . . . . 14
8	1, 7	ax-mp 5	. . . . . . . . . . . . 13
9		hashcl 13147	. . . . . . . . . . . . . . . 16
10	9	ad2antrr 762	. . . . . . . . . . . . . . 15 $/\$ $/\$
11		hashcl 13147	. . . . . . . . . . . . . . . 16
12	11	ad2antlr 763	. . . . . . . . . . . . . . 15 $/\$ $/\$
13		simpr 477	. . . . . . . . . . . . . . 15 $/\$ $/\$
14		nn0sub2 11438	. . . . . . . . . . . . . . 15 $/\$ $/\$
15	10, 12, 13, 14	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$
16		hashfz1 13134	. . . . . . . . . . . . . 14
17	15, 16	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$
18	8, 17	syl5eq 2668	. . . . . . . . . . . 12 $/\$ $/\$
19	6, 18	oveq12d 6668	. . . . . . . . . . 11 $/\$ $/\$
20	9	nn0cnd 11353	. . . . . . . . . . . . 13
21	11	nn0cnd 11353	. . . . . . . . . . . . 13
22		pncan3 10289	. . . . . . . . . . . . 13 $/\$
23	20, 21, 22	syl2an 494	. . . . . . . . . . . 12 $/\$
24	23	adantr 481	. . . . . . . . . . 11 $/\$ $/\$
25	19, 24	eqtrd 2656	. . . . . . . . . 10 $/\$ $/\$
26		ficardom 8787	. . . . . . . . . . . 12
27	26	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$
28	4	hashgadd 13166	. . . . . . . . . . 11 $/\$
29	27, 3, 28	sylancl 694	. . . . . . . . . 10 $/\$ $/\$
30	4	hashgval 13120	. . . . . . . . . . 11
31	30	ad2antlr 763	. . . . . . . . . 10 $/\$ $/\$
32	25, 29, 31	3eqtr4d 2666	. . . . . . . . 9 $/\$ $/\$
33	32	fveq2d 6195	. . . . . . . 8 $/\$ $/\$
34	4	hashgf1o 12770	. . . . . . . . 9
35		nnacl 7691	. . . . . . . . . 10 $/\$
36	27, 3, 35	sylancl 694	. . . . . . . . 9 $/\$ $/\$
37		f1ocnvfv1 6532	. . . . . . . . 9 $/\$
38	34, 36, 37	sylancr 695	. . . . . . . 8 $/\$ $/\$
39		ficardom 8787	. . . . . . . . . 10
40	39	ad2antlr 763	. . . . . . . . 9 $/\$ $/\$
41		f1ocnvfv1 6532	. . . . . . . . 9 $/\$
42	34, 40, 41	sylancr 695	. . . . . . . 8 $/\$ $/\$
43	33, 38, 42	3eqtr3d 2664	. . . . . . 7 $/\$ $/\$
44		oveq2 6658	. . . . . . . . 9
45	44	eqeq1d 2624	. . . . . . . 8
46	45	rspcev 3309	. . . . . . 7 $/\$
47	3, 43, 46	sylancr 695	. . . . . 6 $/\$ $/\$
48	47	ex 450	. . . . 5 $/\$
49		cardnn 8789	. . . . . . . . . 10
50	49	adantl 482	. . . . . . . . 9 $/\$ $/\$
51	50	oveq2d 6666	. . . . . . . 8 $/\$ $/\$
52	51	eqeq1d 2624	. . . . . . 7 $/\$ $/\$
53		fveq2 6191	. . . . . . . 8
54		nnfi 8153	. . . . . . . . 9
55		ficardom 8787	. . . . . . . . . . . . . 14
56	4	hashgadd 13166	. . . . . . . . . . . . . 14 $/\$
57	26, 55, 56	syl2an 494	. . . . . . . . . . . . 13 $/\$
58	4	hashgval 13120	. . . . . . . . . . . . . 14
59	5, 58	oveqan12d 6669	. . . . . . . . . . . . 13 $/\$
60	57, 59	eqtrd 2656	. . . . . . . . . . . 12 $/\$
61	60	adantlr 751	. . . . . . . . . . 11 $/\$ $/\$
62	30	ad2antlr 763	. . . . . . . . . . 11 $/\$ $/\$
63	61, 62	eqeq12d 2637	. . . . . . . . . 10 $/\$ $/\$
64		hashcl 13147	. . . . . . . . . . . . . . 15
65	64	nn0ge0d 11354	. . . . . . . . . . . . . 14
66	65	adantl 482	. . . . . . . . . . . . 13 $/\$
67	9	nn0red 11352	. . . . . . . . . . . . . 14
68	64	nn0red 11352	. . . . . . . . . . . . . 14
69		addge01 10538	. . . . . . . . . . . . . 14 $/\$
70	67, 68, 69	syl2an 494	. . . . . . . . . . . . 13 $/\$
71	66, 70	mpbid 222	. . . . . . . . . . . 12 $/\$
72	71	adantlr 751	. . . . . . . . . . 11 $/\$ $/\$
73		breq2 4657	. . . . . . . . . . 11
74	72, 73	syl5ibcom 235	. . . . . . . . . 10 $/\$ $/\$
75	63, 74	sylbid 230	. . . . . . . . 9 $/\$ $/\$
76	54, 75	sylan2 491	. . . . . . . 8 $/\$ $/\$
77	53, 76	syl5 34	. . . . . . 7 $/\$ $/\$
78	52, 77	sylbird 250	. . . . . 6 $/\$ $/\$
79	78	rexlimdva 3031	. . . . 5 $/\$
80	48, 79	impbid 202	. . . 4 $/\$
81		nnawordex 7717	. . . . 5 $/\$
82	26, 39, 81	syl2an 494	. . . 4 $/\$
83		finnum 8774	. . . . 5
84		finnum 8774	. . . . 5
85		carddom2 8803	. . . . 5 $/\$
86	83, 84, 85	syl2an 494	. . . 4 $/\$
87	80, 82, 86	3bitr2d 296	. . 3 $/\$
88	87	adantlr 751	. 2 $/\$ $/\$
89		hashxrcl 13148	. . . . . 6
90	89	ad2antrr 762	. . . . 5 $/\$ $/\$
91		pnfge 11964	. . . . 5
92	90, 91	syl 17	. . . 4 $/\$ $/\$
93		hashinf 13122	. . . . 5 $/\$
94	93	adantll 750	. . . 4 $/\$ $/\$
95	92, 94	breqtrrd 4681	. . 3 $/\$ $/\$
96		isinffi 8818	. . . . . 6 $/\$
97	96	ancoms 469	. . . . 5 $/\$
98	97	adantlr 751	. . . 4 $/\$ $/\$
99		brdomg 7965	. . . . 5
100	99	ad2antlr 763	. . . 4 $/\$ $/\$
101	98, 100	mpbird 247	. . 3 $/\$ $/\$
102	95, 101	2thd 255	. 2 $/\$ $/\$
103	88, 102	pm2.61dan 832	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wex 1704

wcel 1990

wrex 2913

cvv 3200

wss 3574 class class class wbr 4653

cmpt 4729

ccnv 5113

cdm 5114

cres 5116

wf1 5885

wf1o 5887

cfv 5888 (class class class)co 6650

com 7065

crdg 7505

coa 7557

cdom 7953

cfn 7955

ccrd 8761

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cpnf 10071

cxr 10073

cle 10075

cmin 10266

cn0 11292

cfz 12326

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-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-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-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-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118

This theorem is referenced by: hashdomi 13169 hashsdom 13170 hashun2 13172 hashss 13197 hashsslei 13213 hashfun 13224 hashf1 13241 hashge3el3dif 13268 isercoll 14398 phicl2 15473 phibnd 15476 prmreclem2 15621 prmreclem3 15622 4sqlem11 15659 vdwlem11 15695 ramub2 15718 0ram 15724 ram0 15726 sylow1lem4 18016 pgpssslw 18029 fislw 18040 znfld 19909 znidomb 19910 fta1blem 23928 birthdaylem3 24680 basellem4 24810 ppiwordi 24888 musum 24917 ppiub 24929 chpub 24945 lgsqrlem4 25074 upgrex 25987 sizusglecusg 26359 derangenlem 31153 subfaclefac 31158 erdsze2lem1 31185 snmlff 31311 idomsubgmo 37776 aacllem 42547

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