Theorem cantnff 8571

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 6201	. . . 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 4793	. . 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 2622	. . . 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 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 ) ) )
8		cantnfs.s	. . . . 5 |- S = dom ( A CNF B )
9	4, 5, 6	cantnfdm 8561	. . . . 5 |- ( ph -> dom ( A CNF B ) = { g e. ( A ^m B ) | g finSupp (/) } )
10	8, 9	syl5eq 2668	. . . 4 |- ( ph -> S = { g e. ( A ^m B ) | g finSupp (/) } )
11	10	mpteq1d 4738	. . 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 2659	. 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 481	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> A e. On )
14	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> B e. On )
15		eqid 2622	. . . . . . . 8 |- OrdIso ( _E , ( x supp (/) ) ) = OrdIso ( _E , ( x supp (/) ) )
16		simpr 477	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> x e. S )
17		eqid 2622	. . . . . . . 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 8565	. . . . . . 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 481	. . . . . 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 6678	. . . . . . . . . . 11 |- ( x supp (/) ) e. _V
21	8, 13, 14, 15, 16	cantnfcl 8564	. . . . . . . . . . . 12 |- ( ( ph /\ x e. S ) -> ( _E We ( x supp (/) ) /\ dom OrdIso ( _E , ( x supp (/) ) ) e. om ) )
22	21	simpld 475	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> _E We ( x supp (/) ) )
23	15	oien 8443	. . . . . . . . . . 11 |- ( ( ( x supp (/) ) e. _V /\ _E We ( x supp (/) ) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ ( x supp (/) ) )
24	20, 22, 23	sylancr 695	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ ( x supp (/) ) )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ ( x supp (/) ) )
26		suppssdm 7308	. . . . . . . . . . 11 |- ( x supp (/) ) C_ dom x
27	8, 5, 6	cantnfs 8563	. . . . . . . . . . . . 13 |- ( ph -> ( x e. S <-> ( x : B --> A /\ x finSupp (/) ) ) )
28	27	simprbda 653	. . . . . . . . . . . 12 |- ( ( ph /\ x e. S ) -> x : B --> A )
29		fdm 6051	. . . . . . . . . . . 12 |- ( x : B --> A -> dom x = B )
30	28, 29	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> dom x = B )
31	26, 30	syl5sseq 3653	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( x supp (/) ) C_ B )
32		feq3 6028	. . . . . . . . . . . . . 14 |- ( A = (/) -> ( x : B --> A <-> x : B --> (/) ) )
33	28, 32	syl5ibcom 235	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. S ) -> ( A = (/) -> x : B --> (/) ) )
34	33	imp 445	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> x : B --> (/) )
35		f00 6087	. . . . . . . . . . . 12 |- ( x : B --> (/) <-> ( x = (/) /\ B = (/) ) )
36	34, 35	sylib 208	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( x = (/) /\ B = (/) ) )
37	36	simprd 479	. . . . . . . . . 10 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> B = (/) )
38		sseq0 3975	. . . . . . . . . 10 |- ( ( ( x supp (/) ) C_ B /\ B = (/) ) -> ( x supp (/) ) = (/) )
39	31, 37, 38	syl2an2r 876	. . . . . . . . 9 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( x supp (/) ) = (/) )
40	25, 39	breqtrd 4679	. . . . . . . 8 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) ~~ (/) )
41		en0 8019	. . . . . . . 8 |- ( dom OrdIso ( _E , ( x supp (/) ) ) ~~ (/) <-> dom OrdIso ( _E , ( x supp (/) ) ) = (/) )
42	40, 41	sylib 208	. . . . . . 7 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> dom OrdIso ( _E , ( x supp (/) ) ) = (/) )
43	42	fveq2d 6195	. . . . . 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 ) ) , (/) ) ` (/) ) )
44		0ex 4790	. . . . . . 7 |- (/) e. _V
45	17	seqom0g 7551	. . . . . . 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 ) ) , (/) ) ` (/) ) = (/) )
46	44, 45	mp1i 13	. . . . . 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 ) ) , (/) ) ` (/) ) = (/) )
47	19, 43, 46	3eqtrd 2660	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) = (/) )
48		el1o 7579	. . . . 5 |- ( ( ( A CNF B ) ` x ) e. 1o <-> ( ( A CNF B ) ` x ) = (/) )
49	47, 48	sylibr 224	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) e. 1o )
50	37	oveq2d 6666	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o B ) = ( A ^o (/) ) )
51	13	adantr 481	. . . . . 6 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> A e. On )
52		oe0 7602	. . . . . 6 |- ( A e. On -> ( A ^o (/) ) = 1o )
53	51, 52	syl 17	. . . . 5 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o (/) ) = 1o )
54	50, 53	eqtrd 2656	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( A ^o B ) = 1o )
55	49, 54	eleqtrrd 2704	. . 3 |- ( ( ( ph /\ x e. S ) /\ A = (/) ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
56	13	adantr 481	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> A e. On )
57	14	adantr 481	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> B e. On )
58	16	adantr 481	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> x e. S )
59		on0eln0 5780	. . . . . 6 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
60	13, 59	syl 17	. . . . 5 |- ( ( ph /\ x e. S ) -> ( (/) e. A <-> A =/= (/) ) )
61	60	biimpar 502	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> (/) e. A )
62	31	adantr 481	. . . 4 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> ( x supp (/) ) C_ B )
63	8, 56, 57, 58, 61, 57, 62	cantnflt2 8570	. . 3 |- ( ( ( ph /\ x e. S ) /\ A =/= (/) ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
64	55, 63	pm2.61dane 2881	. 2 |- ( ( ph /\ x e. S ) -> ( ( A CNF B ) ` x ) e. ( A ^o B ) )
65	3, 12, 64	fmpt2d 6393	1 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   [_csb 3533   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   We wwe 5072   dom cdm 5114   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   supp csupp 7295  seq_𝜔cseqom 7542   1oc1o 7553   +o coa 7557   .o comu 7558   ^o coe 7559   ^m cmap 7857   ~~ cen 7952   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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cantnfp1  8578  cantnflem1  8586  cantnflem3  8588  cantnflem4  8589  cantnf  8590