Theorem erdszelem10 31182

Description: Lemma for erdsze 31184. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypotheses

Ref	Expression
erdsze.n	|- ( ph -> N e. NN )
erdsze.f	|- ( ph -> F : ( 1 ... N ) -1-1-> RR )
erdszelem.i	|- I = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
erdszelem.j	|- J = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , `' < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
erdszelem.t	|- T = ( n e. ( 1 ... N ) |-> <. ( I ` n ) , ( J ` n ) >. )
erdszelem.r	|- ( ph -> R e. NN )
erdszelem.s	|- ( ph -> S e. NN )
erdszelem.m	|- ( ph -> ( ( R - 1 ) x. ( S - 1 ) ) < N )

Assertion

Ref	Expression
erdszelem10	|- ( ph -> E. m e. ( 1 ... N ) ( -. ( I ` m ) e. ( 1 ... ( R - 1 ) ) \/ -. ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) )

Distinct variable groups:   x, y   m, n, x, y, F   n, I, x, y   n, J, x, y   R, m, x, y   m, N, n, x, y   ph, m, n, x, y   S, m, x, y   T, m

Allowed substitution hints:   R( n)   S( n)   T( x, y, n)   I( m)   J( m)

Proof of Theorem erdszelem10

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 12771	. . . . . . . 8 |- ( 1 ... ( R - 1 ) ) e. Fin
2		fzfi 12771	. . . . . . . 8 |- ( 1 ... ( S - 1 ) ) e. Fin
3		xpfi 8231	. . . . . . . 8 |- ( ( ( 1 ... ( R - 1 ) ) e. Fin /\ ( 1 ... ( S - 1 ) ) e. Fin ) -> ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) e. Fin )
4	1, 2, 3	mp2an 708	. . . . . . 7 |- ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) e. Fin
5		ssdomg 8001	. . . . . . 7 |- ( ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) e. Fin -> ( ran T C_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) -> ran T ~<_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
6	4, 5	ax-mp 5	. . . . . 6 |- ( ran T C_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) -> ran T ~<_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) )
7		domnsym 8086	. . . . . 6 |- ( ran T ~<_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) -> -. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ran T )
8	6, 7	syl 17	. . . . 5 |- ( ran T C_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) -> -. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ran T )
9		erdszelem.m	. . . . . . . 8 |- ( ph -> ( ( R - 1 ) x. ( S - 1 ) ) < N )
10		hashxp 13221	. . . . . . . . . 10 |- ( ( ( 1 ... ( R - 1 ) ) e. Fin /\ ( 1 ... ( S - 1 ) ) e. Fin ) -> ( # ` ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) = ( ( # ` ( 1 ... ( R - 1 ) ) ) x. ( # ` ( 1 ... ( S - 1 ) ) ) ) )
11	1, 2, 10	mp2an 708	. . . . . . . . 9 |- ( # ` ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) = ( ( # ` ( 1 ... ( R - 1 ) ) ) x. ( # ` ( 1 ... ( S - 1 ) ) ) )
12		erdszelem.r	. . . . . . . . . . 11 |- ( ph -> R e. NN )
13		nnm1nn0 11334	. . . . . . . . . . 11 |- ( R e. NN -> ( R - 1 ) e. NN0 )
14		hashfz1 13134	. . . . . . . . . . 11 |- ( ( R - 1 ) e. NN0 -> ( # ` ( 1 ... ( R - 1 ) ) ) = ( R - 1 ) )
15	12, 13, 14	3syl 18	. . . . . . . . . 10 |- ( ph -> ( # ` ( 1 ... ( R - 1 ) ) ) = ( R - 1 ) )
16		erdszelem.s	. . . . . . . . . . 11 |- ( ph -> S e. NN )
17		nnm1nn0 11334	. . . . . . . . . . 11 |- ( S e. NN -> ( S - 1 ) e. NN0 )
18		hashfz1 13134	. . . . . . . . . . 11 |- ( ( S - 1 ) e. NN0 -> ( # ` ( 1 ... ( S - 1 ) ) ) = ( S - 1 ) )
19	16, 17, 18	3syl 18	. . . . . . . . . 10 |- ( ph -> ( # ` ( 1 ... ( S - 1 ) ) ) = ( S - 1 ) )
20	15, 19	oveq12d 6668	. . . . . . . . 9 |- ( ph -> ( ( # ` ( 1 ... ( R - 1 ) ) ) x. ( # ` ( 1 ... ( S - 1 ) ) ) ) = ( ( R - 1 ) x. ( S - 1 ) ) )
21	11, 20	syl5eq 2668	. . . . . . . 8 |- ( ph -> ( # ` ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) = ( ( R - 1 ) x. ( S - 1 ) ) )
22		erdsze.n	. . . . . . . . . 10 |- ( ph -> N e. NN )
23	22	nnnn0d 11351	. . . . . . . . 9 |- ( ph -> N e. NN0 )
24		hashfz1 13134	. . . . . . . . 9 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
25	23, 24	syl 17	. . . . . . . 8 |- ( ph -> ( # ` ( 1 ... N ) ) = N )
26	9, 21, 25	3brtr4d 4685	. . . . . . 7 |- ( ph -> ( # ` ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) < ( # ` ( 1 ... N ) ) )
27		fzfid 12772	. . . . . . . 8 |- ( ph -> ( 1 ... N ) e. Fin )
28		hashsdom 13170	. . . . . . . 8 |- ( ( ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( # ` ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) < ( # ` ( 1 ... N ) ) <-> ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ( 1 ... N ) ) )
29	4, 27, 28	sylancr 695	. . . . . . 7 |- ( ph -> ( ( # ` ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) < ( # ` ( 1 ... N ) ) <-> ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ( 1 ... N ) ) )
30	26, 29	mpbid 222	. . . . . 6 |- ( ph -> ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ( 1 ... N ) )
31		erdsze.f	. . . . . . . 8 |- ( ph -> F : ( 1 ... N ) -1-1-> RR )
32		erdszelem.i	. . . . . . . 8 |- I = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
33		erdszelem.j	. . . . . . . 8 |- J = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , `' < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
34		erdszelem.t	. . . . . . . 8 |- T = ( n e. ( 1 ... N ) |-> <. ( I ` n ) , ( J ` n ) >. )
35	22, 31, 32, 33, 34	erdszelem9 31181	. . . . . . 7 |- ( ph -> T : ( 1 ... N ) -1-1-> ( NN X. NN ) )
36		f1f1orn 6148	. . . . . . 7 |- ( T : ( 1 ... N ) -1-1-> ( NN X. NN ) -> T : ( 1 ... N ) -1-1-onto-> ran T )
37		ovex 6678	. . . . . . . 8 |- ( 1 ... N ) e. _V
38	37	f1oen 7976	. . . . . . 7 |- ( T : ( 1 ... N ) -1-1-onto-> ran T -> ( 1 ... N ) ~~ ran T )
39	35, 36, 38	3syl 18	. . . . . 6 |- ( ph -> ( 1 ... N ) ~~ ran T )
40		sdomentr 8094	. . . . . 6 |- ( ( ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ( 1 ... N ) /\ ( 1 ... N ) ~~ ran T ) -> ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ran T )
41	30, 39, 40	syl2anc 693	. . . . 5 |- ( ph -> ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ~< ran T )
42	8, 41	nsyl3 133	. . . 4 |- ( ph -> -. ran T C_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) )
43		nss 3663	. . . . 5 |- ( -. ran T C_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> E. s ( s e. ran T /\ -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
44		df-rex 2918	. . . . 5 |- ( E. s e. ran T -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> E. s ( s e. ran T /\ -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
45	43, 44	bitr4i 267	. . . 4 |- ( -. ran T C_ ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> E. s e. ran T -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) )
46	42, 45	sylib 208	. . 3 |- ( ph -> E. s e. ran T -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) )
47		f1fn 6102	. . . 4 |- ( T : ( 1 ... N ) -1-1-> ( NN X. NN ) -> T Fn ( 1 ... N ) )
48		eleq1 2689	. . . . . 6 |- ( s = ( T ` m ) -> ( s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
49	48	notbid 308	. . . . 5 |- ( s = ( T ` m ) -> ( -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
50	49	rexrn 6361	. . . 4 |- ( T Fn ( 1 ... N ) -> ( E. s e. ran T -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> E. m e. ( 1 ... N ) -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
51	35, 47, 50	3syl 18	. . 3 |- ( ph -> ( E. s e. ran T -. s e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> E. m e. ( 1 ... N ) -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
52	46, 51	mpbid 222	. 2 |- ( ph -> E. m e. ( 1 ... N ) -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) )
53		fveq2 6191	. . . . . . . . . 10 |- ( n = m -> ( I ` n ) = ( I ` m ) )
54		fveq2 6191	. . . . . . . . . 10 |- ( n = m -> ( J ` n ) = ( J ` m ) )
55	53, 54	opeq12d 4410	. . . . . . . . 9 |- ( n = m -> <. ( I ` n ) , ( J ` n ) >. = <. ( I ` m ) , ( J ` m ) >. )
56		opex 4932	. . . . . . . . 9 |- <. ( I ` m ) , ( J ` m ) >. e. _V
57	55, 34, 56	fvmpt 6282	. . . . . . . 8 |- ( m e. ( 1 ... N ) -> ( T ` m ) = <. ( I ` m ) , ( J ` m ) >. )
58	57	adantl 482	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... N ) ) -> ( T ` m ) = <. ( I ` m ) , ( J ` m ) >. )
59	58	eleq1d 2686	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... N ) ) -> ( ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> <. ( I ` m ) , ( J ` m ) >. e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) ) )
60		opelxp 5146	. . . . . 6 |- ( <. ( I ` m ) , ( J ` m ) >. e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> ( ( I ` m ) e. ( 1 ... ( R - 1 ) ) /\ ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) )
61	59, 60	syl6bb 276	. . . . 5 |- ( ( ph /\ m e. ( 1 ... N ) ) -> ( ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> ( ( I ` m ) e. ( 1 ... ( R - 1 ) ) /\ ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) ) )
62	61	notbid 308	. . . 4 |- ( ( ph /\ m e. ( 1 ... N ) ) -> ( -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> -. ( ( I ` m ) e. ( 1 ... ( R - 1 ) ) /\ ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) ) )
63		ianor 509	. . . 4 |- ( -. ( ( I ` m ) e. ( 1 ... ( R - 1 ) ) /\ ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) <-> ( -. ( I ` m ) e. ( 1 ... ( R - 1 ) ) \/ -. ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) )
64	62, 63	syl6bb 276	. . 3 |- ( ( ph /\ m e. ( 1 ... N ) ) -> ( -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> ( -. ( I ` m ) e. ( 1 ... ( R - 1 ) ) \/ -. ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) ) )
65	64	rexbidva 3049	. 2 |- ( ph -> ( E. m e. ( 1 ... N ) -. ( T ` m ) e. ( ( 1 ... ( R - 1 ) ) X. ( 1 ... ( S - 1 ) ) ) <-> E. m e. ( 1 ... N ) ( -. ( I ` m ) e. ( 1 ... ( R - 1 ) ) \/ -. ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) ) )
66	52, 65	mpbid 222	1 |- ( ph -> E. m e. ( 1 ... N ) ( -. ( I ` m ) e. ( 1 ... ( R - 1 ) ) \/ -. ( J ` m ) e. ( 1 ... ( S - 1 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   ~Pcpw 4158   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   supcsup 8346   RRcr 9935   1c1 9937   x. cmul 9941   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 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-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  ax-pre-sup 10014

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-isom 5897  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  erdszelem11  31183