Theorem sdc 33540

Description: Strong dependent choice. Suppose we may choose an element of A such that property ps holds, and suppose that if we have already chosen the first k elements (represented here by a function from 1 ... k to A), we may choose another element so that all k + 1 elements taken together have property ps. Then there exists an infinite sequence of elements of A such that the first n terms of this sequence satisfy ps for all n. This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)

Hypotheses

Ref	Expression
sdc.1	|- Z = ( ZZ>= ` M )
sdc.2	|- ( g = ( f |` ( M ... n ) ) -> ( ps <-> ch ) )
sdc.3	|- ( n = M -> ( ps <-> ta ) )
sdc.4	|- ( n = k -> ( ps <-> th ) )
sdc.5	|- ( ( g = h /\ n = ( k + 1 ) ) -> ( ps <-> si ) )
sdc.6	|- ( ph -> A e. V )
sdc.7	|- ( ph -> M e. ZZ )
sdc.8	|- ( ph -> E. g ( g : { M } --> A /\ ta ) )
sdc.9	|- ( ( ph /\ k e. Z ) -> ( ( g : ( M ... k ) --> A /\ th ) -> E. h ( h : ( M ... ( k + 1 ) ) --> A /\ g = ( h |` ( M ... k ) ) /\ si ) ) )

Assertion

Ref	Expression
sdc	|- ( ph -> E. f ( f : Z --> A /\ A. n e. Z ch ) )

Distinct variable groups:   f, g, h, k, n, A   f, M, g, h, k, n   ch, g   ps, f, h, k   si, f, g, n   ph, n   th, n   h, V   ta, h, k, n   f, Z, g, h, k, n   ph, g, h, k

Allowed substitution hints:   ph( f)   ps( g, n)   ch( f, h, k, n)   th( f, g, h, k)   ta( f, g)   si( h, k)   V( f, g, k, n)

Proof of Theorem sdc

Dummy variables j x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdc.1	. 2 |- Z = ( ZZ>= ` M )
2		sdc.2	. 2 |- ( g = ( f |` ( M ... n ) ) -> ( ps <-> ch ) )
3		sdc.3	. 2 |- ( n = M -> ( ps <-> ta ) )
4		sdc.4	. 2 |- ( n = k -> ( ps <-> th ) )
5		sdc.5	. 2 |- ( ( g = h /\ n = ( k + 1 ) ) -> ( ps <-> si ) )
6		sdc.6	. 2 |- ( ph -> A e. V )
7		sdc.7	. 2 |- ( ph -> M e. ZZ )
8		sdc.8	. 2 |- ( ph -> E. g ( g : { M } --> A /\ ta ) )
9		sdc.9	. 2 |- ( ( ph /\ k e. Z ) -> ( ( g : ( M ... k ) --> A /\ th ) -> E. h ( h : ( M ... ( k + 1 ) ) --> A /\ g = ( h |` ( M ... k ) ) /\ si ) ) )
10		eqid 2622	. 2 |- { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) } = { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) }
11		eqid 2622	. . . 4 |- Z = Z
12		oveq2 6658	. . . . . . . 8 |- ( n = k -> ( M ... n ) = ( M ... k ) )
13	12	feq2d 6031	. . . . . . 7 |- ( n = k -> ( g : ( M ... n ) --> A <-> g : ( M ... k ) --> A ) )
14	13, 4	anbi12d 747	. . . . . 6 |- ( n = k -> ( ( g : ( M ... n ) --> A /\ ps ) <-> ( g : ( M ... k ) --> A /\ th ) ) )
15	14	cbvrexv 3172	. . . . 5 |- ( E. n e. Z ( g : ( M ... n ) --> A /\ ps ) <-> E. k e. Z ( g : ( M ... k ) --> A /\ th ) )
16	15	abbii 2739	. . . 4 |- { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) } = { g | E. k e. Z ( g : ( M ... k ) --> A /\ th ) }
17		eqid 2622	. . . 4 |- { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } = { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) }
18	11, 16, 17	mpt2eq123i 6718	. . 3 |- ( j e. Z , f e. { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) } |-> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } ) = ( j e. Z , f e. { g | E. k e. Z ( g : ( M ... k ) --> A /\ th ) } |-> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } )
19		eqidd 2623	. . . 4 |- ( j = y -> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } = { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } )
20		eqeq1 2626	. . . . . . 7 |- ( f = x -> ( f = ( h |` ( M ... k ) ) <-> x = ( h |` ( M ... k ) ) ) )
21	20	3anbi2d 1404	. . . . . 6 |- ( f = x -> ( ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) <-> ( h : ( M ... ( k + 1 ) ) --> A /\ x = ( h |` ( M ... k ) ) /\ si ) ) )
22	21	rexbidv 3052	. . . . 5 |- ( f = x -> ( E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) <-> E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ x = ( h |` ( M ... k ) ) /\ si ) ) )
23	22	abbidv 2741	. . . 4 |- ( f = x -> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } = { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ x = ( h |` ( M ... k ) ) /\ si ) } )
24	19, 23	cbvmpt2v 6735	. . 3 |- ( j e. Z , f e. { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) } |-> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } ) = ( y e. Z , x e. { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) } |-> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ x = ( h |` ( M ... k ) ) /\ si ) } )
25	18, 24	eqtr3i 2646	. 2 |- ( j e. Z , f e. { g | E. k e. Z ( g : ( M ... k ) --> A /\ th ) } |-> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ f = ( h |` ( M ... k ) ) /\ si ) } ) = ( y e. Z , x e. { g | E. n e. Z ( g : ( M ... n ) --> A /\ ps ) } |-> { h | E. k e. Z ( h : ( M ... ( k + 1 ) ) --> A /\ x = ( h |` ( M ... k ) ) /\ si ) } )
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25	sdclem1 33539	1 |- ( ph -> E. f ( f : Z --> A /\ A. n e. Z ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   {csn 4177   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

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-inf2 8538  ax-dc 9268  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-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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by: (None)