Theorem erdszelem7 29932

Description: Lemma for erdsze 29937. (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 12529	. . . 4 |- # : _V --> ( NN0 u. { +oo } )
2		ffun 5736	. . . 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 29930	. . . 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 675	. . 3 |- ( ph -> ( K ` A ) e. ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) )
11		fvelima 5922	. . 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 670	. 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 2453	. . . . . 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 29926	. . . . 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 1054	. . . . . . . . 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 11804	. . . . . . . . . . 11 |- ( A e. ( 1 ... N ) -> N e. ( ZZ>= ` A ) )
17		fzss2 11845	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` A ) -> ( 1 ... A ) C_ ( 1 ... N ) )
18	4, 16, 17	3syl 18	. . . . . . . . . 10 |- ( ph -> ( 1 ... A ) C_ ( 1 ... N ) )
19	18	adantr 467	. . . . . . . . 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 3444	. . . . . . . 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 3960	. . . . . . . 8 |- ( s e. ~P ( 1 ... N ) <-> s C_ ( 1 ... N ) )
22	20, 21	sylibr 216	. . . . . . 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 29931	. . . . . . . . . . . . . . 15 |- ( ph -> K : ( 1 ... N ) --> NN )
25	24, 4	ffvelrnd 6028	. . . . . . . . . . . . . 14 |- ( ph -> ( K ` A ) e. NN )
26		nnuz 11201	. . . . . . . . . . . . . 14 |- NN = ( ZZ>= ` 1 )
27	25, 26	syl6eleq 2541	. . . . . . . . . . . . 13 |- ( ph -> ( K ` A ) e. ( ZZ>= ` 1 ) )
28		erdszelem7.r	. . . . . . . . . . . . . 14 |- ( ph -> R e. NN )
29		nnz 10966	. . . . . . . . . . . . . 14 |- ( R e. NN -> R e. ZZ )
30		peano2zm 10987	. . . . . . . . . . . . . 14 |- ( R e. ZZ -> ( R - 1 ) e. ZZ )
31	28, 29, 30	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> ( R - 1 ) e. ZZ )
32		elfz5 11799	. . . . . . . . . . . . 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 667	. . . . . . . . . . . 12 |- ( ph -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) <_ ( R - 1 ) ) )
34		nnltlem1 11010	. . . . . . . . . . . . 13 |- ( ( ( K ` A ) e. NN /\ R e. NN ) -> ( ( K ` A ) < R <-> ( K ` A ) <_ ( R - 1 ) ) )
35	25, 28, 34	syl2anc 667	. . . . . . . . . . . 12 |- ( ph -> ( ( K ` A ) < R <-> ( K ` A ) <_ ( R - 1 ) ) )
36	33, 35	bitr4d 260	. . . . . . . . . . 11 |- ( ph -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) < R ) )
37	23, 36	mtbid 302	. . . . . . . . . 10 |- ( ph -> -. ( K ` A ) < R )
38	28	nnred 10631	. . . . . . . . . . 11 |- ( ph -> R e. RR )
39	13	erdszelem2 29927	. . . . . . . . . . . . . 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 464	. . . . . . . . . . . . 13 |- ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ NN
41		nnssre 10620	. . . . . . . . . . . . 13 |- NN C_ RR
42	40, 41	sstri 3443	. . . . . . . . . . . 12 |- ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ RR
43	42, 10	sseldi 3432	. . . . . . . . . . 11 |- ( ph -> ( K ` A ) e. RR )
44	38, 43	lenltd 9786	. . . . . . . . . 10 |- ( ph -> ( R <_ ( K ` A ) <-> -. ( K ` A ) < R ) )
45	37, 44	mpbird 236	. . . . . . . . 9 |- ( ph -> R <_ ( K ` A ) )
46	45	adantr 467	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> R <_ ( K ` A ) )
47		simprr 767	. . . . . . . 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 4432	. . . . . . 7 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> R <_ ( # ` s ) )
49		simprl2 1055	. . . . . . 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 538	. . . . . 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 620	. . . . 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 478	. . . 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 608	. . 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 2860	. 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 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   E.wrex 2740   {crab 2743   _Vcvv 3047   u. cun 3404   C_ wss 3406   ~Pcpw 3953   {csn 3970   class class class wbr 4405   |-> cmpt 4464   Or wor 4757   |` cres 4839   "cima 4840   Fun wfun 5579   -->wf 5581   -1-1->wf1 5582   ` cfv 5585   Isom wiso 5586  (class class class)co 6295   Fincfn 7574   supcsup 7959   RRcr 9543   1c1 9545   +oocpnf 9677   < clt 9680   <_ cle 9681   - cmin 9865   NNcn 10616   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791   #chash 12522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523

This theorem is referenced by:  erdszelem11  29936