Theorem erdszelem7 28473

Description: Lemma for erdsze 28478. (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.k	|- K = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
erdszelem.o	|- O Or RR
erdszelem.a	|- ( ph -> A e. ( 1 ... N ) )
erdszelem7.r	|- ( ph -> R e. NN )
erdszelem7.m	|- ( ph -> -. ( K ` A ) e. ( 1 ... ( R - 1 ) ) )

Assertion

Ref	Expression
erdszelem7	|- ( ph -> E. s e. ~P ( 1 ... N ) ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) )

Distinct variable groups:   x, y, s, F   K, s   A, s, x, y   O, s, x, y   R, s, x, y   N, s, x, y   ph, s, x, y

Allowed substitution hints:   K( x, y)

Proof of Theorem erdszelem7

Step	Hyp	Ref	Expression
1		hashf 12392	. . . 4 |- # : _V --> ( NN0 u. { +oo } )
2		ffun 5739	. . . 4 |- ( # : _V --> ( NN0 u. { +oo } ) -> Fun # )
3	1, 2	ax-mp 5	. . 3 |- Fun #
4		erdszelem.a	. . . 4 |- ( ph -> A e. ( 1 ... N ) )
5		erdsze.n	. . . . 5 |- ( ph -> N e. NN )
6		erdsze.f	. . . . 5 |- ( ph -> F : ( 1 ... N ) -1-1-> RR )
7		erdszelem.k	. . . . 5 |- K = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
8		erdszelem.o	. . . . 5 |- O Or RR
9	5, 6, 7, 8	erdszelem5 28471	. . . 4 |- ( ( ph /\ A e. ( 1 ... N ) ) -> ( K ` A ) e. ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) )
10	4, 9	mpdan 668	. . 3 |- ( ph -> ( K ` A ) e. ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) )
11		fvelima 5926	. . 3 |- ( ( Fun # /\ ( K ` A ) e. ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) ) -> E. s e. { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ( # ` s ) = ( K ` A ) )
12	3, 10, 11	sylancr 663	. 2 |- ( ph -> E. s e. { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ( # ` s ) = ( K ` A ) )
13		eqid 2467	. . . . . 6 |- { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } = { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) }
14	13	erdszelem1 28467	. . . . 5 |- ( s e. { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } <-> ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) )
15		simprl1 1041	. . . . . . . . 9 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> s C_ ( 1 ... A ) )
16		elfzuz3 11697	. . . . . . . . . . 11 |- ( A e. ( 1 ... N ) -> N e. ( ZZ>= ` A ) )
17		fzss2 11735	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` A ) -> ( 1 ... A ) C_ ( 1 ... N ) )
18	4, 16, 17	3syl 20	. . . . . . . . . 10 |- ( ph -> ( 1 ... A ) C_ ( 1 ... N ) )
19	18	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> ( 1 ... A ) C_ ( 1 ... N ) )
20	15, 19	sstrd 3519	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> s C_ ( 1 ... N ) )
21		selpw 4023	. . . . . . . 8 |- ( s e. ~P ( 1 ... N ) <-> s C_ ( 1 ... N ) )
22	20, 21	sylibr 212	. . . . . . 7 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> s e. ~P ( 1 ... N ) )
23		erdszelem7.m	. . . . . . . . . . 11 |- ( ph -> -. ( K ` A ) e. ( 1 ... ( R - 1 ) ) )
24	5, 6, 7, 8	erdszelem6 28472	. . . . . . . . . . . . . . 15 |- ( ph -> K : ( 1 ... N ) --> NN )
25	24, 4	ffvelrnd 6033	. . . . . . . . . . . . . 14 |- ( ph -> ( K ` A ) e. NN )
26		nnuz 11129	. . . . . . . . . . . . . 14 |- NN = ( ZZ>= ` 1 )
27	25, 26	syl6eleq 2565	. . . . . . . . . . . . 13 |- ( ph -> ( K ` A ) e. ( ZZ>= ` 1 ) )
28		erdszelem7.r	. . . . . . . . . . . . . 14 |- ( ph -> R e. NN )
29		nnz 10898	. . . . . . . . . . . . . 14 |- ( R e. NN -> R e. ZZ )
30		peano2zm 10918	. . . . . . . . . . . . . 14 |- ( R e. ZZ -> ( R - 1 ) e. ZZ )
31	28, 29, 30	3syl 20	. . . . . . . . . . . . 13 |- ( ph -> ( R - 1 ) e. ZZ )
32		elfz5 11692	. . . . . . . . . . . . 13 |- ( ( ( K ` A ) e. ( ZZ>= ` 1 ) /\ ( R - 1 ) e. ZZ ) -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) <_ ( R - 1 ) ) )
33	27, 31, 32	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) <_ ( R - 1 ) ) )
34		nnltlem1 10940	. . . . . . . . . . . . 13 |- ( ( ( K ` A ) e. NN /\ R e. NN ) -> ( ( K ` A ) < R <-> ( K ` A ) <_ ( R - 1 ) ) )
35	25, 28, 34	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> ( ( K ` A ) < R <-> ( K ` A ) <_ ( R - 1 ) ) )
36	33, 35	bitr4d 256	. . . . . . . . . . 11 |- ( ph -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) < R ) )
37	23, 36	mtbid 300	. . . . . . . . . 10 |- ( ph -> -. ( K ` A ) < R )
38	28	nnred 10563	. . . . . . . . . . 11 |- ( ph -> R e. RR )
39	13	erdszelem2 28468	. . . . . . . . . . . . . 14 |- ( ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) e. Fin /\ ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ NN )
40	39	simpri 462	. . . . . . . . . . . . 13 |- ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ NN
41		nnssre 10552	. . . . . . . . . . . . 13 |- NN C_ RR
42	40, 41	sstri 3518	. . . . . . . . . . . 12 |- ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ RR
43	42, 10	sseldi 3507	. . . . . . . . . . 11 |- ( ph -> ( K ` A ) e. RR )
44	38, 43	lenltd 9742	. . . . . . . . . 10 |- ( ph -> ( R <_ ( K ` A ) <-> -. ( K ` A ) < R ) )
45	37, 44	mpbird 232	. . . . . . . . 9 |- ( ph -> R <_ ( K ` A ) )
46	45	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> R <_ ( K ` A ) )
47		simprr 756	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> ( # ` s ) = ( K ` A ) )
48	46, 47	breqtrrd 4479	. . . . . . 7 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> R <_ ( # ` s ) )
49		simprl2 1042	. . . . . . 7 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> ( F |` s ) Isom < , O ( s , ( F " s ) ) )
50	22, 48, 49	jca32 535	. . . . . 6 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> ( s e. ~P ( 1 ... N ) /\ ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) ) )
51	50	expr 615	. . . . 5 |- ( ( ph /\ ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) ) -> ( ( # ` s ) = ( K ` A ) -> ( s e. ~P ( 1 ... N ) /\ ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) ) ) )
52	14, 51	sylan2b 475	. . . 4 |- ( ( ph /\ s e. { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) -> ( ( # ` s ) = ( K ` A ) -> ( s e. ~P ( 1 ... N ) /\ ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) ) ) )
53	52	expimpd 603	. . 3 |- ( ph -> ( ( s e. { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } /\ ( # ` s ) = ( K ` A ) ) -> ( s e. ~P ( 1 ... N ) /\ ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) ) ) )
54	53	reximdv2 2938	. 2 |- ( ph -> ( E. s e. { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ( # ` s ) = ( K ` A ) -> E. s e. ~P ( 1 ... N ) ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) ) )
55	12, 54	mpd 15	1 |- ( ph -> E. s e. ~P ( 1 ... N ) ( R <_ ( # ` s ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   E.wrex 2818   {crab 2821   _Vcvv 3118   u. cun 3479   C_ wss 3481   ~Pcpw 4016   {csn 4033   class class class wbr 4453   |-> cmpt 4511   Or wor 4805   |` cres 5007   "cima 5008   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591   ` cfv 5594   Isom wiso 5595  (class class class)co 6295   Fincfn 7528   supcsup 7912   RRcr 9503   1c1 9505   +oocpnf 9637   < clt 9640   <_ cle 9641   - cmin 9817   NNcn 10548   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   #chash 12385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386

This theorem is referenced by:  erdszelem11  28477