Theorem cantnfval 8565

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
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnfcl.g	|- G = OrdIso ( _E , ( F supp (/) ) )
cantnfcl.f	|- ( ph -> F e. S )
cantnfval.h	|- H = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) , (/) )

Assertion

Ref	Expression
cantnfval	|- ( ph -> ( ( A CNF B ) ` F ) = ( H ` dom G ) )

Distinct variable groups:   z, k, B   A, k, z   k, F, z   S, k, z   k, G, z   ph, k, z

Allowed substitution hints:   H( z, k)

Proof of Theorem cantnfval

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- { g e. ( A ^m B ) | g finSupp (/) } = { g e. ( A ^m B ) | g finSupp (/) }
2		cantnfs.a	. . . 4 |- ( ph -> A e. On )
3		cantnfs.b	. . . 4 |- ( ph -> B e. On )
4	1, 2, 3	cantnffval 8560	. . 3 |- ( ph -> ( A CNF B ) = ( f e. { g e. ( A ^m B ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) )
5	4	fveq1d 6193	. 2 |- ( ph -> ( ( A CNF B ) ` F ) = ( ( f e. { g e. ( A ^m B ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) ` F ) )
6		cantnfcl.f	. . . 4 |- ( ph -> F e. S )
7		cantnfs.s	. . . . 5 |- S = dom ( A CNF B )
8	1, 2, 3	cantnfdm 8561	. . . . 5 |- ( ph -> dom ( A CNF B ) = { g e. ( A ^m B ) | g finSupp (/) } )
9	7, 8	syl5eq 2668	. . . 4 |- ( ph -> S = { g e. ( A ^m B ) | g finSupp (/) } )
10	6, 9	eleqtrd 2703	. . 3 |- ( ph -> F e. { g e. ( A ^m B ) | g finSupp (/) } )
11		ovex 6678	. . . . . 6 |- ( f supp (/) ) e. _V
12		eqid 2622	. . . . . . 7 |- OrdIso ( _E , ( f supp (/) ) ) = OrdIso ( _E , ( f supp (/) ) )
13	12	oiexg 8440	. . . . . 6 |- ( ( f supp (/) ) e. _V -> OrdIso ( _E , ( f supp (/) ) ) e. _V )
14	11, 13	mp1i 13	. . . . 5 |- ( f = F -> OrdIso ( _E , ( f supp (/) ) ) e. _V )
15		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> h = OrdIso ( _E , ( f supp (/) ) ) )
16		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( f = F -> ( f supp (/) ) = ( F supp (/) ) )
17	16	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( f supp (/) ) = ( F supp (/) ) )
18		oieq2 8418	. . . . . . . . . . . . . . . 16 |- ( ( f supp (/) ) = ( F supp (/) ) -> OrdIso ( _E , ( f supp (/) ) ) = OrdIso ( _E , ( F supp (/) ) ) )
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> OrdIso ( _E , ( f supp (/) ) ) = OrdIso ( _E , ( F supp (/) ) ) )
20	15, 19	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> h = OrdIso ( _E , ( F supp (/) ) ) )
21		cantnfcl.g	. . . . . . . . . . . . . 14 |- G = OrdIso ( _E , ( F supp (/) ) )
22	20, 21	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> h = G )
23	22	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( h ` k ) = ( G ` k ) )
24	23	oveq2d 6666	. . . . . . . . . . 11 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( A ^o ( h ` k ) ) = ( A ^o ( G ` k ) ) )
25		simpl 473	. . . . . . . . . . . 12 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> f = F )
26	25, 23	fveq12d 6197	. . . . . . . . . . 11 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( f ` ( h ` k ) ) = ( F ` ( G ` k ) ) )
27	24, 26	oveq12d 6668	. . . . . . . . . 10 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) = ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) )
28	27	oveq1d 6665	. . . . . . . . 9 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) = ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) )
29	28	mpt2eq3dv 6721	. . . . . . . 8 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) = ( k e. _V , z e. _V |-> ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) )
30		eqid 2622	. . . . . . . 8 |- (/) = (/)
31		seqomeq12 7549	. . . . . . . 8 |- ( ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) = ( k e. _V , z e. _V |-> ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) /\ (/) = (/) ) -> seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) , (/) ) )
32	29, 30, 31	sylancl 694	. . . . . . 7 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) , (/) ) )
33		cantnfval.h	. . . . . . 7 |- H = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) , (/) )
34	32, 33	syl6eqr 2674	. . . . . 6 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) = H )
35	22	dmeqd 5326	. . . . . 6 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> dom h = dom G )
36	34, 35	fveq12d 6197	. . . . 5 |- ( ( f = F /\ h = OrdIso ( _E , ( f supp (/) ) ) ) -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) = ( H ` dom G ) )
37	14, 36	csbied 3560	. . . 4 |- ( f = F -> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) = ( H ` dom G ) )
38		eqid 2622	. . . 4 |- ( f e. { g e. ( A ^m B ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) = ( f e. { g e. ( A ^m B ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) )
39		fvex 6201	. . . 4 |- ( H ` dom G ) e. _V
40	37, 38, 39	fvmpt 6282	. . 3 |- ( F e. { g e. ( A ^m B ) | g finSupp (/) } -> ( ( f e. { g e. ( A ^m B ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) ` F ) = ( H ` dom G ) )
41	10, 40	syl 17	. 2 |- ( ph -> ( ( f e. { g e. ( A ^m B ) | g finSupp (/) } |-> [_OrdIso ( _E , ( f supp (/) ) ) / h ]_ (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) ` F ) = ( H ` dom G ) )
42	5, 41	eqtrd 2656	1 |- ( ph -> ( ( A CNF B ) ` F ) = ( H ` dom G ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cantnfval2  8566  cantnfle  8568  cantnflt2  8570  cantnff  8571  cantnf0  8572  cantnfp1lem3  8577  cantnflem1  8586  cantnf  8590  cnfcom2  8599