Theorem cantnffval 8560

Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)

Hypotheses

Ref	Expression
cantnffval.s	|- S = { g e. ( A ^m B ) | g finSupp (/) }
cantnffval.a	|- ( ph -> A e. On )
cantnffval.b	|- ( ph -> B e. On )

Assertion

Ref	Expression
cantnffval	|- ( ph -> ( A CNF B ) = ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )

Distinct variable groups:   f, g, h, k, z, A   B, f, g, h, k, z   S, f

Allowed substitution hints:   ph( z, f, g, h, k)   S( z, g, h, k)

Proof of Theorem cantnffval

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

Step	Hyp	Ref	Expression
1		cantnffval.a	. 2 |- ( ph -> A e. On )
2		cantnffval.b	. 2 |- ( ph -> B e. On )
3		oveq12 6659	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( x ^m y ) = ( A ^m B ) )
4	3	rabeqdv 3194	. . . . 5 |- ( ( x = A /\ y = B ) -> { g e. ( x ^m y ) | g finSupp (/) } = { g e. ( A ^m B ) | g finSupp (/) } )
5		cantnffval.s	. . . . 5 |- S = { g e. ( A ^m B ) | g finSupp (/) }
6	4, 5	syl6eqr 2674	. . . 4 |- ( ( x = A /\ y = B ) -> { g e. ( x ^m y ) | g finSupp (/) } = S )
7		simp1l 1085	. . . . . . . . . . 11 |- ( ( ( x = A /\ y = B ) /\ k e. _V /\ z e. _V ) -> x = A )
8	7	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( x = A /\ y = B ) /\ k e. _V /\ z e. _V ) -> ( x ^o ( h ` k ) ) = ( A ^o ( h ` k ) ) )
9	8	oveq1d 6665	. . . . . . . . 9 |- ( ( ( x = A /\ y = B ) /\ k e. _V /\ z e. _V ) -> ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) = ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) )
10	9	oveq1d 6665	. . . . . . . 8 |- ( ( ( x = A /\ y = B ) /\ k e. _V /\ z e. _V ) -> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) = ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) )
11	10	mpt2eq3dva 6719	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) = ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) )
12		eqid 2622	. . . . . . 7 |- (/) = (/)
13		seqomeq12 7549	. . . . . . 7 |- ( ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) = ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) /\ (/) = (/) ) -> seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) )
14	11, 12, 13	sylancl 694	. . . . . 6 |- ( ( x = A /\ y = B ) -> seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) )
15	14	fveq1d 6193	. . . . 5 |- ( ( x = A /\ y = B ) -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) )
16	15	csbeq2dv 3992	. . . 4 |- ( ( x = A /\ y = B ) -> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) = [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) )
17	6, 16	mpteq12dv 4733	. . 3 |- ( ( x = A /\ y = B ) -> ( f e. { g e. ( x ^m y ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) = ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
18		df-cnf 8559	. . 3 |- CNF = ( x e. On , y e. On |-> ( f e. { g e. ( x ^m y ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
19		ovex 6678	. . . . 5 |- ( A ^m B ) e. _V
20	5, 19	rabex2 4815	. . . 4 |- S e. _V
21	20	mptex 6486	. . 3 |- ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) e. _V
22	17, 18, 21	ovmpt2a 6791	. 2 |- ( ( A e. On /\ B e. On ) -> ( A CNF B ) = ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
23	1, 2, 22	syl2anc 693	1 |- ( ph -> ( A CNF B ) = ( f e. S |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [_csb 3533   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   dom cdm 5114   Oncon0 5723   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295  seq_𝜔cseqom 7542   +o coa 7557   .o comu 7558   ^o coe 7559   ^m cmap 7857   finSupp cfsupp 8275  OrdIsocoi 8414   CNF ccnf 8558

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-cnf 8559

This theorem is referenced by:  cantnfdm  8561  cantnfval  8565  cantnff  8571