Theorem cantnff 7997

Description: The CNF function is a function from finitely supported functions from B to A, to the ordinal exponential A ^o B. (Contributed by Mario Carneiro, 28-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )

Assertion

Ref	Expression
cantnff	|- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )

Proof of Theorem cantnff

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

Step	Hyp	Ref	Expression
1		fvex 5812	. . . 4 |- (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) e. _V
2	1	csbex 4536	. . 3 |- [_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
3	2	a1i 11	. 2 |- ( ( ph /\ 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 )
4		eqid 2454	. . . 4 |- { g e. ( A ^m B ) | g finSupp (/) } = { g e. ( A ^m B ) | g finSupp (/) }
5		cantnfs.a	. . . 4 |- ( ph -> A e. On )
6		cantnfs.b	. . . 4 |- ( ph -> B e. On )
7	4, 5, 6	cantnffval 7984	. . 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 ) ) )
8		cantnfs.s	. . . . 5 |- S = dom ( A CNF B )
9	4, 5, 6	cantnfdm 7985	. . . . 5 |- ( ph -> dom ( A CNF B ) = { g e. ( A ^m B ) | g finSupp (/) } )
10	8, 9	syl5eq 2507	. . . 4 |- ( ph -> S = { g e. ( A ^m B ) | g finSupp (/) } )
11	10	mpteq1d 4484	. . 3 |- ( ph -> ( 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 ) ) = ( 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 ) ) )
12	7, 11	eqtr4d 2498	. 2 |- ( 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 ) ) )
13	5	adantr 465	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> A e. On )
14	6	adantr 465	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> B e. On )
15		eqid 2454	. . . . . . . 8 |- OrdIso ( _E , ( x supp (/) ) ) = OrdIso ( _E , ( x supp (/) ) )
16		simpr 461	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> x e. S )
17		eqid 2454	. . . . . . . 8 |- seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) )
18	8, 13, 14, 15, 16, 17	cantnfval 7991	. . . . . . 7 |- ( ( ph /\ x e. S ) -> ( ( A CNF B ) ` x ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( x supp (/) ) ) ) )
19	18	adantr 465	. . . . . 6 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( x supp (/) ) ) ) )
20		ovex 6228	. . . . . . . . . . 11 |- ( x supp (/) ) e. _V
21	8, 13, 14, 15, 16	cantnfcl 7990	. . . . . . . . . . . 12 |- ( ( ph /\ x e. S ) -> ( _E We ( x supp (/) ) /\ dom OrdIso ( _E , ( x supp (/) ) ) e. om ) )
22	21	simpld 459	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> _E We ( x supp (/) ) )
23	15	oien 7867	. . . . . . . . . . 11 |- ( ( ( x supp (/) ) e. _V /\ _E We ( x supp (/) ) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ ( x supp (/) ) )
24	20, 22, 23	sylancr 663	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ ( x supp (/) ) )
25	24	adantr 465	. . . . . . . . 9 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ ( x supp (/) ) )
26		suppssdm 6816	. . . . . . . . . . . 12 |- ( x supp (/) ) C_ dom x
27	8, 5, 6	cantnfs 7989	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. S <-> ( x : B --> A /\ x finSupp (/) ) ) )
28	27	simprbda 623	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. S ) -> x : B --> A )
29		fdm 5674	. . . . . . . . . . . . 13 |- ( x : B --> A -> dom x = B )
30	28, 29	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ x e. S ) -> dom x = B )
31	26, 30	syl5sseq 3515	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> ( x supp (/) ) C_ B )
32	31	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( x supp (/) ) C_ B )
33		feq3 5655	. . . . . . . . . . . . . 14 |- ( A = (/) -> ( x : B --> A <-> x : B --> (/) ) )
34	28, 33	syl5ibcom 220	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. S ) -> ( A = (/) -> x : B --> (/) ) )
35	34	imp 429	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> x : B --> (/) )
36		f00 5704	. . . . . . . . . . . 12 |- ( x : B --> (/) <-> ( x = (/) /\ B = (/) ) )
37	35, 36	sylib 196	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( x = (/) /\ B = (/) ) )
38	37	simprd 463	. . . . . . . . . 10 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> B = (/) )
39		sseq0 3780	. . . . . . . . . 10 |- ( ( ( x supp (/) ) C_ B /\ B = (/) ) -> ( x supp (/) ) = (/) )
40	32, 38, 39	syl2anc 661	. . . . . . . . 9 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( x supp (/) ) = (/) )
41	25, 40	breqtrd 4427	. . . . . . . 8 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ (/) )
42		en0 7485	. . . . . . . 8 |- ( dom OrdIso ( _E , ( x supp (/) ) ) ~~ (/) <-> dom OrdIso ( _E , ( x supp (/) ) ) = (/) )
43	41, 42	sylib 196	. . . . . . 7 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) = (/) )
44	43	fveq2d 5806	. . . . . 6 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( x supp (/) ) ) ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` (/) ) )
45		0ex 4533	. . . . . . 7 |- (/) e. _V
46	17	seqom0g 7024	. . . . . . 7 |- ( (/) e. _V -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` (/) ) = (/) )
47	45, 46	mp1i 12	. . . . . 6 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) .o ( x ` (OrdIso ( _E , ( x supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` (/) ) = (/) )
48	19, 44, 47	3eqtrd 2499	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) = (/) )
49		el1o 7052	. . . . 5 |- ( ( ( A CNF B ) ` x ) e. 1o <-> ( ( A CNF B ) ` x ) = (/) )
50	48, 49	sylibr 212	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) e. 1o )
51	38	oveq2d 6219	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o B ) = ( A ^o (/) ) )
52	13	adantr 465	. . . . . 6 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> A e. On )
53		oe0 7075	. . . . . 6 |- ( A e. On -> ( A ^o (/) ) = 1o )
54	52, 53	syl 16	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o (/) ) = 1o )
55	51, 54	eqtrd 2495	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o B ) = 1o )
56	50, 55	eleqtrrd 2545	. . 3 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
57	13	adantr 465	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> A e. On )
58	14	adantr 465	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> B e. On )
59	16	adantr 465	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> x e. S )
60		on0eln0 4885	. . . . . 6 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
61	13, 60	syl 16	. . . . 5 |- ( ( ph /\ x e. S ) -> ( (/) e. A <-> A =/= (/) ) )
62	61	biimpar 485	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> (/) e. A )
63	31	adantr 465	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> ( x supp (/) ) C_ B )
64	8, 57, 58, 59, 62, 58, 63	cantnflt2 7996	. . 3 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
65	56, 64	pm2.61dane 2770	. 2 |- ( ( ph /\ x e. S ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
66	3, 12, 65	fmpt2d 5985	1 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   {crab 2803   _Vcvv 3078   [_csb 3398   C_ wss 3439   (/)c0 3748   class class class wbr 4403   |-> cmpt 4461   _E cep 4741   We wwe 4789   Oncon0 4830   dom cdm 4951   -->wf 5525   ` cfv 5529  (class class class)co 6203   |-> cmpt2 6205   omcom 6589   supp csupp 6803  seq_𝜔cseqom 7015   1oc1o 7026   +o coa 7030   .o comu 7031   ^o coe 7032   ^m cmap 7327   ~~ cen 7420   finSupp cfsupp 7734  OrdIsocoi 7838   CNF ccnf 7982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7839  df-cnf 7983

This theorem is referenced by:  cantnfp1  8004  cantnflem1  8012  cantnflem3  8014  cantnflem4  8015  cantnf  8016  cantnfp1OLD  8030  cantnflem1OLD  8035  cantnflem3OLD  8036  cantnflem4OLD  8037  cantnfOLD  8038