Theorem cantnf0 8572

Description: The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnf0.a	|- ( ph -> (/) e. A )

Assertion

Ref	Expression
cantnf0	|- ( ph -> ( ( A CNF B ) ` ( B X. { (/) } ) ) = (/) )

Proof of Theorem cantnf0

Dummy variables k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . 3 |- S = dom ( A CNF B )
2		cantnfs.a	. . 3 |- ( ph -> A e. On )
3		cantnfs.b	. . 3 |- ( ph -> B e. On )
4		eqid 2622	. . 3 |- OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) = OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) )
5		cantnf0.a	. . . . 5 |- ( ph -> (/) e. A )
6		fconst6g 6094	. . . . 5 |- ( (/) e. A -> ( B X. { (/) } ) : B --> A )
7	5, 6	syl 17	. . . 4 |- ( ph -> ( B X. { (/) } ) : B --> A )
8	3, 5	fczfsuppd 8293	. . . 4 |- ( ph -> ( B X. { (/) } ) finSupp (/) )
9	1, 2, 3	cantnfs 8563	. . . 4 |- ( ph -> ( ( B X. { (/) } ) e. S <-> ( ( B X. { (/) } ) : B --> A /\ ( B X. { (/) } ) finSupp (/) ) ) )
10	7, 8, 9	mpbir2and 957	. . 3 |- ( ph -> ( B X. { (/) } ) e. S )
11		eqid 2622	. . 3 |- seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) )
12	1, 2, 3, 4, 10, 11	cantnfval 8565	. 2 |- ( ph -> ( ( A CNF B ) ` ( B X. { (/) } ) ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ) )
13		eqidd 2623	. . . . . . 7 |- ( ph -> ( B X. { (/) } ) = ( B X. { (/) } ) )
14		0ex 4790	. . . . . . . . 9 |- (/) e. _V
15		fnconstg 6093	. . . . . . . . 9 |- ( (/) e. _V -> ( B X. { (/) } ) Fn B )
16	14, 15	mp1i 13	. . . . . . . 8 |- ( ph -> ( B X. { (/) } ) Fn B )
17	14	a1i 11	. . . . . . . 8 |- ( ph -> (/) e. _V )
18		fnsuppeq0 7323	. . . . . . . 8 |- ( ( ( B X. { (/) } ) Fn B /\ B e. On /\ (/) e. _V ) -> ( ( ( B X. { (/) } ) supp (/) ) = (/) <-> ( B X. { (/) } ) = ( B X. { (/) } ) ) )
19	16, 3, 17, 18	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( ( B X. { (/) } ) supp (/) ) = (/) <-> ( B X. { (/) } ) = ( B X. { (/) } ) ) )
20	13, 19	mpbird 247	. . . . . 6 |- ( ph -> ( ( B X. { (/) } ) supp (/) ) = (/) )
21		oieq2 8418	. . . . . 6 |- ( ( ( B X. { (/) } ) supp (/) ) = (/) -> OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) = OrdIso ( _E , (/) ) )
22	20, 21	syl 17	. . . . 5 |- ( ph -> OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) = OrdIso ( _E , (/) ) )
23	22	dmeqd 5326	. . . 4 |- ( ph -> dom OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) = dom OrdIso ( _E , (/) ) )
24		we0 5109	. . . . . 6 |- _E We (/)
25		eqid 2622	. . . . . . 7 |- OrdIso ( _E , (/) ) = OrdIso ( _E , (/) )
26	25	oien 8443	. . . . . 6 |- ( ( (/) e. _V /\ _E We (/) ) -> dom OrdIso ( _E , (/) ) ~~ (/) )
27	14, 24, 26	mp2an 708	. . . . 5 |- dom OrdIso ( _E , (/) ) ~~ (/)
28		en0 8019	. . . . 5 |- ( dom OrdIso ( _E , (/) ) ~~ (/) <-> dom OrdIso ( _E , (/) ) = (/) )
29	27, 28	mpbi 220	. . . 4 |- dom OrdIso ( _E , (/) ) = (/)
30	23, 29	syl6eq 2672	. . 3 |- ( ph -> dom OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) = (/) )
31	30	fveq2d 6195	. 2 |- ( ph -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` dom OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ) = (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` (/) ) )
32	11	seqom0g 7551	. . 3 |- ( (/) e. _V -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` (/) ) = (/) )
33	14, 32	mp1i 13	. 2 |- ( ph -> (seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) .o ( ( B X. { (/) } ) ` (OrdIso ( _E , ( ( B X. { (/) } ) supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) ` (/) ) = (/) )
34	12, 31, 33	3eqtrd 2660	1 |- ( ph -> ( ( A CNF B ) ` ( B X. { (/) } ) ) = (/) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   class class class wbr 4653   _E cep 5028   We wwe 5072   X. cxp 5112   dom cdm 5114   Oncon0 5723   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295  seq_𝜔cseqom 7542   +o coa 7557   .o comu 7558   ^o coe 7559   ~~ 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-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-map 7859  df-en 7956  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cnfcom2lem  8598