Theorem erdszelem7 24888

Description: Lemma for erdsze 24893. (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 11630	. . . 4 |- # : _V --> ( NN0 u. { +oo } )
2		ffun 5596	. . . 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 24886	. . . 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 651	. . 3 |- ( ph -> ( K ` A ) e. ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) )
11		fvelima 5781	. . 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 646	. 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 2438	. . . . . 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 24882	. . . . 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 1003	. . . . . . . . 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 11061	. . . . . . . . . . 11 |- ( A e. ( 1 ... N ) -> N e. ( ZZ>= ` A ) )
17		fzss2 11097	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` A ) -> ( 1 ... A ) C_ ( 1 ... N ) )
18	4, 16, 17	3syl 19	. . . . . . . . . 10 |- ( ph -> ( 1 ... A ) C_ ( 1 ... N ) )
19	18	adantr 453	. . . . . . . . 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 3360	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> s C_ ( 1 ... N ) )
21		vex 2961	. . . . . . . . 9 |- s e. _V
22	21	elpw 3807	. . . . . . . 8 |- ( s e. ~P ( 1 ... N ) <-> s C_ ( 1 ... N ) )
23	20, 22	sylibr 205	. . . . . . 7 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> s e. ~P ( 1 ... N ) )
24		erdszelem7.m	. . . . . . . . . . 11 |- ( ph -> -. ( K ` A ) e. ( 1 ... ( R - 1 ) ) )
25	5, 6, 7, 8	erdszelem6 24887	. . . . . . . . . . . . . . 15 |- ( ph -> K : ( 1 ... N ) --> NN )
26	25, 4	ffvelrnd 5874	. . . . . . . . . . . . . 14 |- ( ph -> ( K ` A ) e. NN )
27		nnuz 10526	. . . . . . . . . . . . . 14 |- NN = ( ZZ>= ` 1 )
28	26, 27	syl6eleq 2528	. . . . . . . . . . . . 13 |- ( ph -> ( K ` A ) e. ( ZZ>= ` 1 ) )
29		erdszelem7.r	. . . . . . . . . . . . . 14 |- ( ph -> R e. NN )
30		nnz 10308	. . . . . . . . . . . . . 14 |- ( R e. NN -> R e. ZZ )
31		peano2zm 10325	. . . . . . . . . . . . . 14 |- ( R e. ZZ -> ( R - 1 ) e. ZZ )
32	29, 30, 31	3syl 19	. . . . . . . . . . . . 13 |- ( ph -> ( R - 1 ) e. ZZ )
33		elfz5 11056	. . . . . . . . . . . . 13 |- ( ( ( K ` A ) e. ( ZZ>= ` 1 ) /\ ( R - 1 ) e. ZZ ) -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) <_ ( R - 1 ) ) )
34	28, 32, 33	syl2anc 644	. . . . . . . . . . . 12 |- ( ph -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) <_ ( R - 1 ) ) )
35		nnltlem1 10344	. . . . . . . . . . . . 13 |- ( ( ( K ` A ) e. NN /\ R e. NN ) -> ( ( K ` A ) < R <-> ( K ` A ) <_ ( R - 1 ) ) )
36	26, 29, 35	syl2anc 644	. . . . . . . . . . . 12 |- ( ph -> ( ( K ` A ) < R <-> ( K ` A ) <_ ( R - 1 ) ) )
37	34, 36	bitr4d 249	. . . . . . . . . . 11 |- ( ph -> ( ( K ` A ) e. ( 1 ... ( R - 1 ) ) <-> ( K ` A ) < R ) )
38	24, 37	mtbid 293	. . . . . . . . . 10 |- ( ph -> -. ( K ` A ) < R )
39	29	nnred 10020	. . . . . . . . . . 11 |- ( ph -> R e. RR )
40	13	erdszelem2 24883	. . . . . . . . . . . . . 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 )
41	40	simpri 450	. . . . . . . . . . . . 13 |- ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ NN
42		nnssre 10009	. . . . . . . . . . . . 13 |- NN C_ RR
43	41, 42	sstri 3359	. . . . . . . . . . . 12 |- ( # " { y e. ~P ( 1 ... A ) | ( ( F |` y ) Isom < , O ( y , ( F " y ) ) /\ A e. y ) } ) C_ RR
44	43, 10	sseldi 3348	. . . . . . . . . . 11 |- ( ph -> ( K ` A ) e. RR )
45	39, 44	lenltd 9224	. . . . . . . . . 10 |- ( ph -> ( R <_ ( K ` A ) <-> -. ( K ` A ) < R ) )
46	38, 45	mpbird 225	. . . . . . . . 9 |- ( ph -> R <_ ( K ` A ) )
47	46	adantr 453	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> R <_ ( K ` A ) )
48		simprr 735	. . . . . . . 8 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> ( # ` s ) = ( K ` A ) )
49	47, 48	breqtrrd 4241	. . . . . . 7 |- ( ( ph /\ ( ( s C_ ( 1 ... A ) /\ ( F |` s ) Isom < , O ( s , ( F " s ) ) /\ A e. s ) /\ ( # ` s ) = ( K ` A ) ) ) -> R <_ ( # ` s ) )
50		simprl2 1004	. . . . . . 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 ) ) )
51	23, 49, 50	jca32 523	. . . . . 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 ) ) ) ) )
52	51	expr 600	. . . . 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 ) ) ) ) ) )
53	14, 52	sylan2b 463	. . . 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 ) ) ) ) ) )
54	53	expimpd 588	. . 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 ) ) ) ) ) )
55	54	reximdv2 2817	. 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 ) ) ) ) )
56	12, 55	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 178   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   E.wrex 2708   {crab 2711   _Vcvv 2958   u. cun 3320   C_ wss 3322   ~Pcpw 3801   {csn 3816   class class class wbr 4215   e. cmpt 4269   Or wor 4505   |` cres 4883   "cima 4884   Fun wfun 5451   -->wf 5453   -1-1->wf1 5454   ` cfv 5457   Isom wiso 5458  (class class class)co 6084   Fincfn 7112   supcsup 7448   RRcr 8994   1c1 8996   +oocpnf 9122   < clt 9125   <_ cle 9126   - cmin 9296   NNcn 10005   NN0cn0 10226   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048   #chash 11623

This theorem is referenced by:  erdszelem11  24892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-hash 11624