Theorem ackbij1lem5 9046

Description: Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Assertion

Ref	Expression
ackbij1lem5	|- ( A e. om -> ( card ` ~P suc A ) = ( ( card ` ~P A ) +o ( card ` ~P A ) ) )

Proof of Theorem ackbij1lem5

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 5790	. . . . 5 |- ( a = A -> suc a = suc A )
2	1	pweqd 4163	. . . 4 |- ( a = A -> ~P suc a = ~P suc A )
3	2	fveq2d 6195	. . 3 |- ( a = A -> ( card ` ~P suc a ) = ( card ` ~P suc A ) )
4		pweq 4161	. . . . 5 |- ( a = A -> ~P a = ~P A )
5	4	fveq2d 6195	. . . 4 |- ( a = A -> ( card ` ~P a ) = ( card ` ~P A ) )
6	5, 5	oveq12d 6668	. . 3 |- ( a = A -> ( ( card ` ~P a ) +o ( card ` ~P a ) ) = ( ( card ` ~P A ) +o ( card ` ~P A ) ) )
7	3, 6	eqeq12d 2637	. 2 |- ( a = A -> ( ( card ` ~P suc a ) = ( ( card ` ~P a ) +o ( card ` ~P a ) ) <-> ( card ` ~P suc A ) = ( ( card ` ~P A ) +o ( card ` ~P A ) ) ) )
8		vex 3203	. . . . . . . . 9 |- a e. _V
9	8	sucex 7011	. . . . . . . 8 |- suc a e. _V
10	9	pw2en 8067	. . . . . . 7 |- ~P suc a ~~ ( 2o ^m suc a )
11		df-suc 5729	. . . . . . . . . 10 |- suc a = ( a u. { a } )
12	11	oveq2i 6661	. . . . . . . . 9 |- ( 2o ^m suc a ) = ( 2o ^m ( a u. { a } ) )
13		nnord 7073	. . . . . . . . . . 11 |- ( a e. om -> Ord a )
14		orddisj 5762	. . . . . . . . . . 11 |- ( Ord a -> ( a i^i { a } ) = (/) )
15		snex 4908	. . . . . . . . . . . 12 |- { a } e. _V
16		2onn 7720	. . . . . . . . . . . . 13 |- 2o e. om
17	16	elexi 3213	. . . . . . . . . . . 12 |- 2o e. _V
18		mapunen 8129	. . . . . . . . . . . . 13 |- ( ( ( a e. _V /\ { a } e. _V /\ 2o e. _V ) /\ ( a i^i { a } ) = (/) ) -> ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) )
19	18	ex 450	. . . . . . . . . . . 12 |- ( ( a e. _V /\ { a } e. _V /\ 2o e. _V ) -> ( ( a i^i { a } ) = (/) -> ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) ) )
20	8, 15, 17, 19	mp3an 1424	. . . . . . . . . . 11 |- ( ( a i^i { a } ) = (/) -> ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) )
21	13, 14, 20	3syl 18	. . . . . . . . . 10 |- ( a e. om -> ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) )
22		ovex 6678	. . . . . . . . . . . 12 |- ( 2o ^m a ) e. _V
23	22	enref 7988	. . . . . . . . . . 11 |- ( 2o ^m a ) ~~ ( 2o ^m a )
24	17, 8	mapsnen 8035	. . . . . . . . . . 11 |- ( 2o ^m { a } ) ~~ 2o
25		xpen 8123	. . . . . . . . . . 11 |- ( ( ( 2o ^m a ) ~~ ( 2o ^m a ) /\ ( 2o ^m { a } ) ~~ 2o ) -> ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) ~~ ( ( 2o ^m a ) X. 2o ) )
26	23, 24, 25	mp2an 708	. . . . . . . . . 10 |- ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) ~~ ( ( 2o ^m a ) X. 2o )
27		entr 8008	. . . . . . . . . 10 |- ( ( ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) /\ ( ( 2o ^m a ) X. ( 2o ^m { a } ) ) ~~ ( ( 2o ^m a ) X. 2o ) ) -> ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. 2o ) )
28	21, 26, 27	sylancl 694	. . . . . . . . 9 |- ( a e. om -> ( 2o ^m ( a u. { a } ) ) ~~ ( ( 2o ^m a ) X. 2o ) )
29	12, 28	syl5eqbr 4688	. . . . . . . 8 |- ( a e. om -> ( 2o ^m suc a ) ~~ ( ( 2o ^m a ) X. 2o ) )
30	8	pw2en 8067	. . . . . . . . . 10 |- ~P a ~~ ( 2o ^m a )
31	17	enref 7988	. . . . . . . . . 10 |- 2o ~~ 2o
32		xpen 8123	. . . . . . . . . 10 |- ( ( ~P a ~~ ( 2o ^m a ) /\ 2o ~~ 2o ) -> ( ~P a X. 2o ) ~~ ( ( 2o ^m a ) X. 2o ) )
33	30, 31, 32	mp2an 708	. . . . . . . . 9 |- ( ~P a X. 2o ) ~~ ( ( 2o ^m a ) X. 2o )
34	33	ensymi 8006	. . . . . . . 8 |- ( ( 2o ^m a ) X. 2o ) ~~ ( ~P a X. 2o )
35		entr 8008	. . . . . . . 8 |- ( ( ( 2o ^m suc a ) ~~ ( ( 2o ^m a ) X. 2o ) /\ ( ( 2o ^m a ) X. 2o ) ~~ ( ~P a X. 2o ) ) -> ( 2o ^m suc a ) ~~ ( ~P a X. 2o ) )
36	29, 34, 35	sylancl 694	. . . . . . 7 |- ( a e. om -> ( 2o ^m suc a ) ~~ ( ~P a X. 2o ) )
37		entr 8008	. . . . . . 7 |- ( ( ~P suc a ~~ ( 2o ^m suc a ) /\ ( 2o ^m suc a ) ~~ ( ~P a X. 2o ) ) -> ~P suc a ~~ ( ~P a X. 2o ) )
38	10, 36, 37	sylancr 695	. . . . . 6 |- ( a e. om -> ~P suc a ~~ ( ~P a X. 2o ) )
39		vpwex 4849	. . . . . . 7 |- ~P a e. _V
40		xp2cda 9002	. . . . . . 7 |- ( ~P a e. _V -> ( ~P a X. 2o ) = ( ~P a +c ~P a ) )
41	39, 40	ax-mp 5	. . . . . 6 |- ( ~P a X. 2o ) = ( ~P a +c ~P a )
42	38, 41	syl6breq 4694	. . . . 5 |- ( a e. om -> ~P suc a ~~ ( ~P a +c ~P a ) )
43		nnfi 8153	. . . . . . . . 9 |- ( a e. om -> a e. Fin )
44		pwfi 8261	. . . . . . . . 9 |- ( a e. Fin <-> ~P a e. Fin )
45	43, 44	sylib 208	. . . . . . . 8 |- ( a e. om -> ~P a e. Fin )
46		ficardid 8788	. . . . . . . 8 |- ( ~P a e. Fin -> ( card ` ~P a ) ~~ ~P a )
47	45, 46	syl 17	. . . . . . 7 |- ( a e. om -> ( card ` ~P a ) ~~ ~P a )
48		cdaen 8995	. . . . . . 7 |- ( ( ( card ` ~P a ) ~~ ~P a /\ ( card ` ~P a ) ~~ ~P a ) -> ( ( card ` ~P a ) +c ( card ` ~P a ) ) ~~ ( ~P a +c ~P a ) )
49	47, 47, 48	syl2anc 693	. . . . . 6 |- ( a e. om -> ( ( card ` ~P a ) +c ( card ` ~P a ) ) ~~ ( ~P a +c ~P a ) )
50	49	ensymd 8007	. . . . 5 |- ( a e. om -> ( ~P a +c ~P a ) ~~ ( ( card ` ~P a ) +c ( card ` ~P a ) ) )
51		entr 8008	. . . . 5 |- ( ( ~P suc a ~~ ( ~P a +c ~P a ) /\ ( ~P a +c ~P a ) ~~ ( ( card ` ~P a ) +c ( card ` ~P a ) ) ) -> ~P suc a ~~ ( ( card ` ~P a ) +c ( card ` ~P a ) ) )
52	42, 50, 51	syl2anc 693	. . . 4 |- ( a e. om -> ~P suc a ~~ ( ( card ` ~P a ) +c ( card ` ~P a ) ) )
53		carden2b 8793	. . . 4 |- ( ~P suc a ~~ ( ( card ` ~P a ) +c ( card ` ~P a ) ) -> ( card ` ~P suc a ) = ( card ` ( ( card ` ~P a ) +c ( card ` ~P a ) ) ) )
54	52, 53	syl 17	. . 3 |- ( a e. om -> ( card ` ~P suc a ) = ( card ` ( ( card ` ~P a ) +c ( card ` ~P a ) ) ) )
55		ficardom 8787	. . . . 5 |- ( ~P a e. Fin -> ( card ` ~P a ) e. om )
56	45, 55	syl 17	. . . 4 |- ( a e. om -> ( card ` ~P a ) e. om )
57		nnacda 9023	. . . 4 |- ( ( ( card ` ~P a ) e. om /\ ( card ` ~P a ) e. om ) -> ( card ` ( ( card ` ~P a ) +c ( card ` ~P a ) ) ) = ( ( card ` ~P a ) +o ( card ` ~P a ) ) )
58	56, 56, 57	syl2anc 693	. . 3 |- ( a e. om -> ( card ` ( ( card ` ~P a ) +c ( card ` ~P a ) ) ) = ( ( card ` ~P a ) +o ( card ` ~P a ) ) )
59	54, 58	eqtrd 2656	. 2 |- ( a e. om -> ( card ` ~P suc a ) = ( ( card ` ~P a ) +o ( card ` ~P a ) ) )
60	7, 59	vtoclga 3272	1 |- ( A e. om -> ( card ` ~P suc A ) = ( ( card ` ~P A ) +o ( card ` ~P A ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   X. cxp 5112   Ord word 5722   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   2oc2o 7554   +o coa 7557   ^m cmap 7857   ~~ cen 7952   Fincfn 7955   cardccrd 8761   +c ccda 8989

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-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-int 4476  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by:  ackbij1lem14  9055