Theorem cantnff 8105

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 5882	. . . 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 4586	. . 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 2467	. . . 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 8092	. . 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 8093	. . . . 5 |- ( ph -> dom ( A CNF B ) = { g e. ( A ^m B ) | g finSupp (/) } )
10	8, 9	syl5eq 2520	. . . 4 |- ( ph -> S = { g e. ( A ^m B ) | g finSupp (/) } )
11	10	mpteq1d 4534	. . 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 2511	. 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 2467	. . . . . . . 8 |- OrdIso ( _E , ( x supp (/) ) ) = OrdIso ( _E , ( x supp (/) ) )
16		simpr 461	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> x e. S )
17		eqid 2467	. . . . . . . 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 8099	. . . . . . 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 6320	. . . . . . . . . . 11 |- ( x supp (/) ) e. _V
21	8, 13, 14, 15, 16	cantnfcl 8098	. . . . . . . . . . . 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 7975	. . . . . . . . . . 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 6926	. . . . . . . . . . . 12 |- ( x supp (/) ) C_ dom x
27	8, 5, 6	cantnfs 8097	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. S <-> ( x : B --> A /\ x finSupp (/) ) ) )
28	27	simprbda 623	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. S ) -> x : B --> A )
29		fdm 5741	. . . . . . . . . . . . 13 |- ( x : B --> A -> dom x = B )
30	28, 29	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ x e. S ) -> dom x = B )
31	26, 30	syl5sseq 3557	. . . . . . . . . . 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 5721	. . . . . . . . . . . . . 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 5773	. . . . . . . . . . . 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 3822	. . . . . . . . . 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 4477	. . . . . . . 8 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ (/) )
42		en0 7590	. . . . . . . 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 5876	. . . . . 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 4583	. . . . . . 7 |- (/) e. _V
46	17	seqom0g 7133	. . . . . . 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 2512	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) = (/) )
49		el1o 7161	. . . . 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 6311	. . . . 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 7184	. . . . . 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 2508	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o B ) = 1o )
56	50, 55	eleqtrrd 2558	. . 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 4939	. . . . . 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 8104	. . 3 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
65	56, 64	pm2.61dane 2785	. 2 |- ( ( ph /\ x e. S ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
66	3, 12, 65	fmpt2d 6062	1 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   {crab 2821   _Vcvv 3118   [_csb 3440   C_ wss 3481   (/)c0 3790   class class class wbr 4453   |-> cmpt 4511   _E cep 4795   We wwe 4843   Oncon0 4884   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295   |-> cmpt2 6297   omcom 6695   supp csupp 6913  seq_𝜔cseqom 7124   1oc1o 7135   +o coa 7139   .o comu 7140   ^o coe 7141   ^m cmap 7432   ~~ cen 7525   finSupp cfsupp 7841  OrdIsocoi 7946   CNF ccnf 8090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-seqom 7125  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-oexp 7148  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-cnf 8091

This theorem is referenced by:  cantnfp1  8112  cantnflem1  8120  cantnflem3  8122  cantnflem4  8123  cantnf  8124  cantnfp1OLD  8138  cantnflem1OLD  8143  cantnflem3OLD  8144  cantnflem4OLD  8145  cantnfOLD  8146