Theorem cantnff 8110

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 4590	. . 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 2457	. . . 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 8097	. . 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 8098	. . . . 5 |- ( ph -> dom ( A CNF B ) = { g e. ( A ^m B ) | g finSupp (/) } )
10	8, 9	syl5eq 2510	. . . 4 |- ( ph -> S = { g e. ( A ^m B ) | g finSupp (/) } )
11	10	mpteq1d 4538	. . 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 2501	. 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 2457	. . . . . . . 8 |- OrdIso ( _E , ( x supp (/) ) ) = OrdIso ( _E , ( x supp (/) ) )
16		simpr 461	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> x e. S )
17		eqid 2457	. . . . . . . 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 8104	. . . . . . 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 6324	. . . . . . . . . . 11 |- ( x supp (/) ) e. _V
21	8, 13, 14, 15, 16	cantnfcl 8103	. . . . . . . . . . . 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 7981	. . . . . . . . . . 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 6930	. . . . . . . . . . . 12 |- ( x supp (/) ) C_ dom x
27	8, 5, 6	cantnfs 8102	. . . . . . . . . . . . . 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 3547	. . . . . . . . . . 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 3826	. . . . . . . . . 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 4480	. . . . . . . 8 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ (/) )
42		en0 7597	. . . . . . . 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 4587	. . . . . . 7 |- (/) e. _V
46	17	seqom0g 7139	. . . . . . 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 2502	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) = (/) )
49		el1o 7167	. . . . 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 6312	. . . . 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 7190	. . . . . 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 2498	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o B ) = 1o )
56	50, 55	eleqtrrd 2548	. . 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 4942	. . . . . 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 8109	. . 3 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
65	56, 64	pm2.61dane 2775	. 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 1395   e. wcel 1819   =/= wne 2652   {crab 2811   _Vcvv 3109   [_csb 3430   C_ wss 3471   (/)c0 3793   class class class wbr 4456   |-> cmpt 4515   _E cep 4798   We wwe 4846   Oncon0 4887   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   omcom 6699   supp csupp 6917  seq_𝜔cseqom 7130   1oc1o 7141   +o coa 7145   .o comu 7146   ^o coe 7147   ^m cmap 7438   ~~ cen 7532   finSupp cfsupp 7847  OrdIsocoi 7952   CNF ccnf 8095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096

This theorem is referenced by:  cantnfp1  8117  cantnflem1  8125  cantnflem3  8127  cantnflem4  8128  cantnf  8129  cantnfp1OLD  8143  cantnflem1OLD  8148  cantnflem3OLD  8149  cantnflem4OLD  8150  cantnfOLD  8151