Theorem mzpcompact2 37315

Description: Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Assertion

Ref	Expression
mzpcompact2	|- ( A e. (mzPoly ` B ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) )

Distinct variable groups:   A, a, b   B, a, b, c

Allowed substitution hint:   A( c)

Proof of Theorem mzpcompact2

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( A e. (mzPoly ` B ) -> B e. _V )
2		fveq2 6191	. . . . 5 |- ( d = B -> (mzPoly ` d ) = (mzPoly ` B ) )
3	2	eleq2d 2687	. . . 4 |- ( d = B -> ( A e. (mzPoly ` d ) <-> A e. (mzPoly ` B ) ) )
4		sseq2 3627	. . . . . 6 |- ( d = B -> ( a C_ d <-> a C_ B ) )
5		oveq2 6658	. . . . . . . 8 |- ( d = B -> ( ZZ ^m d ) = ( ZZ ^m B ) )
6	5	mpteq1d 4738	. . . . . . 7 |- ( d = B -> ( c e. ( ZZ ^m d ) |-> ( b ` ( c |` a ) ) ) = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) )
7	6	eqeq2d 2632	. . . . . 6 |- ( d = B -> ( A = ( c e. ( ZZ ^m d ) |-> ( b ` ( c |` a ) ) ) <-> A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) )
8	4, 7	anbi12d 747	. . . . 5 |- ( d = B -> ( ( a C_ d /\ A = ( c e. ( ZZ ^m d ) |-> ( b ` ( c |` a ) ) ) ) <-> ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) ) )
9	8	2rexbidv 3057	. . . 4 |- ( d = B -> ( E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ d /\ A = ( c e. ( ZZ ^m d ) |-> ( b ` ( c |` a ) ) ) ) <-> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) ) )
10	3, 9	imbi12d 334	. . 3 |- ( d = B -> ( ( A e. (mzPoly ` d ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ d /\ A = ( c e. ( ZZ ^m d ) |-> ( b ` ( c |` a ) ) ) ) ) <-> ( A e. (mzPoly ` B ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) ) ) )
11		vex 3203	. . . 4 |- d e. _V
12	11	mzpcompact2lem 37314	. . 3 |- ( A e. (mzPoly ` d ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ d /\ A = ( c e. ( ZZ ^m d ) |-> ( b ` ( c |` a ) ) ) ) )
13	10, 12	vtoclg 3266	. 2 |- ( B e. _V -> ( A e. (mzPoly ` B ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) ) )
14	1, 13	mpcom 38	1 |- ( A e. (mzPoly ` B ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   ZZcz 11377  mzPolycmzp 37285

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

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-mzpcl 37286  df-mzp 37287

This theorem is referenced by:  eldioph2  37325