Theorem eventfin 4517

Description: If M is even , then so is T_fin M. Theorem X.1.37 of [Rosser] p. 530. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
eventfin	|- (M e. Evenfin -> Tfin M e. Evenfin )

Proof of Theorem eventfin

Dummy variables m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . . . 6 |- (x = M -> (x = (n +o n) <-> M = (n +o n)))
2	1	rexbidv 2635	. . . . 5 |- (x = M -> (E.n e. Nn x = (n +o n) <-> E.n e. Nn M = (n +o n)))
3		neeq1 2524	. . . . 5 |- (x = M -> (x =/= (/) <-> M =/= (/)))
4	2, 3	anbi12d 691	. . . 4 |- (x = M -> ((E.n e. Nn x = (n +o n) /\ x =/= (/)) <-> (E.n e. Nn M = (n +o n) /\ M =/= (/))))
5		df-evenfin 4444	. . . 4 |- Evenfin = {x | (E.n e. Nn x = (n +o n) /\ x =/= (/))}
6	4, 5	elab2g 2987	. . 3 |- (M e. Evenfin -> (M e. Evenfin <-> (E.n e. Nn M = (n +o n) /\ M =/= (/))))
7	6	ibi 232	. 2 |- (M e. Evenfin -> (E.n e. Nn M = (n +o n) /\ M =/= (/)))
8		addceq2 4384	. . . . . . . . . . . 12 |- (n = (/) -> (n +o n) = (n +o (/)))
9		addcnul1 4452	. . . . . . . . . . . 12 |- (n +o (/)) = (/)
10	8, 9	syl6eq 2401	. . . . . . . . . . 11 |- (n = (/) -> (n +o n) = (/))
11	10	necon3i 2555	. . . . . . . . . 10 |- ((n +o n) =/= (/) -> n =/= (/))
12		tfinprop 4489	. . . . . . . . . . 11 |- ((n e. Nn /\ n =/= (/)) -> ( Tfin n e. Nn /\ E.x e. n ~P1 x e. Tfin n))
13	12	simpld 445	. . . . . . . . . 10 |- ((n e. Nn /\ n =/= (/)) -> Tfin n e. Nn )
14	11, 13	sylan2 460	. . . . . . . . 9 |- ((n e. Nn /\ (n +o n) =/= (/)) -> Tfin n e. Nn )
15		tfindi 4496	. . . . . . . . . 10 |- ((n e. Nn /\ n e. Nn /\ (n +o n) =/= (/)) -> Tfin (n +o n) = ( Tfin n +o Tfin n))
16	15	3anidm12 1239	. . . . . . . . 9 |- ((n e. Nn /\ (n +o n) =/= (/)) -> Tfin (n +o n) = ( Tfin n +o Tfin n))
17		addceq12 4385	. . . . . . . . . . . 12 |- ((m = Tfin n /\ m = Tfin n) -> (m +o m) = ( Tfin n +o Tfin n))
18	17	anidms 626	. . . . . . . . . . 11 |- (m = Tfin n -> (m +o m) = ( Tfin n +o Tfin n))
19	18	eqeq2d 2364	. . . . . . . . . 10 |- (m = Tfin n -> ( Tfin (n +o n) = (m +o m) <-> Tfin (n +o n) = ( Tfin n +o Tfin n)))
20	19	rspcev 2955	. . . . . . . . 9 |- (( Tfin n e. Nn /\ Tfin (n +o n) = ( Tfin n +o Tfin n)) -> E.m e. Nn Tfin (n +o n) = (m +o m))
21	14, 16, 20	syl2anc 642	. . . . . . . 8 |- ((n e. Nn /\ (n +o n) =/= (/)) -> E.m e. Nn Tfin (n +o n) = (m +o m))
22		nncaddccl 4419	. . . . . . . . . 10 |- ((n e. Nn /\ n e. Nn ) -> (n +o n) e. Nn )
23	22	anidms 626	. . . . . . . . 9 |- (n e. Nn -> (n +o n) e. Nn )
24		tfinnnul 4490	. . . . . . . . 9 |- (((n +o n) e. Nn /\ (n +o n) =/= (/)) -> Tfin (n +o n) =/= (/))
25	23, 24	sylan 457	. . . . . . . 8 |- ((n e. Nn /\ (n +o n) =/= (/)) -> Tfin (n +o n) =/= (/))
26	21, 25	jca 518	. . . . . . 7 |- ((n e. Nn /\ (n +o n) =/= (/)) -> (E.m e. Nn Tfin (n +o n) = (m +o m) /\ Tfin (n +o n) =/= (/)))
27		tfinex 4485	. . . . . . . 8 |- Tfin (n +o n) e. _V
28		eqeq1 2359	. . . . . . . . . 10 |- (x = Tfin (n +o n) -> (x = (m +o m) <-> Tfin (n +o n) = (m +o m)))
29	28	rexbidv 2635	. . . . . . . . 9 |- (x = Tfin (n +o n) -> (E.m e. Nn x = (m +o m) <-> E.m e. Nn Tfin (n +o n) = (m +o m)))
30		neeq1 2524	. . . . . . . . 9 |- (x = Tfin (n +o n) -> (x =/= (/) <-> Tfin (n +o n) =/= (/)))
31	29, 30	anbi12d 691	. . . . . . . 8 |- (x = Tfin (n +o n) -> ((E.m e. Nn x = (m +o m) /\ x =/= (/)) <-> (E.m e. Nn Tfin (n +o n) = (m +o m) /\ Tfin (n +o n) =/= (/))))
32		df-evenfin 4444	. . . . . . . 8 |- Evenfin = {x | (E.m e. Nn x = (m +o m) /\ x =/= (/))}
33	27, 31, 32	elab2 2988	. . . . . . 7 |- ( Tfin (n +o n) e. Evenfin <-> (E.m e. Nn Tfin (n +o n) = (m +o m) /\ Tfin (n +o n) =/= (/)))
34	26, 33	sylibr 203	. . . . . 6 |- ((n e. Nn /\ (n +o n) =/= (/)) -> Tfin (n +o n) e. Evenfin )
35	34	ex 423	. . . . 5 |- (n e. Nn -> ((n +o n) =/= (/) -> Tfin (n +o n) e. Evenfin ))
36		neeq1 2524	. . . . . . . 8 |- (M = (n +o n) -> (M =/= (/) <-> (n +o n) =/= (/)))
37		tfineq 4488	. . . . . . . . 9 |- (M = (n +o n) -> Tfin M = Tfin (n +o n))
38	37	eleq1d 2419	. . . . . . . 8 |- (M = (n +o n) -> ( Tfin M e. Evenfin <-> Tfin (n +o n) e. Evenfin ))
39	36, 38	imbi12d 311	. . . . . . 7 |- (M = (n +o n) -> ((M =/= (/) -> Tfin M e. Evenfin ) <-> ((n +o n) =/= (/) -> Tfin (n +o n) e. Evenfin )))
40	39	biimprd 214	. . . . . 6 |- (M = (n +o n) -> (((n +o n) =/= (/) -> Tfin (n +o n) e. Evenfin ) -> (M =/= (/) -> Tfin M e. Evenfin )))
41	40	com12 27	. . . . 5 |- (((n +o n) =/= (/) -> Tfin (n +o n) e. Evenfin ) -> (M = (n +o n) -> (M =/= (/) -> Tfin M e. Evenfin )))
42	35, 41	syl 15	. . . 4 |- (n e. Nn -> (M = (n +o n) -> (M =/= (/) -> Tfin M e. Evenfin )))
43	42	rexlimiv 2732	. . 3 |- (E.n e. Nn M = (n +o n) -> (M =/= (/) -> Tfin M e. Evenfin ))
44	43	imp 418	. 2 |- ((E.n e. Nn M = (n +o n) /\ M =/= (/)) -> Tfin M e. Evenfin )
45	7, 44	syl 15	1 |- (M e. Evenfin -> Tfin M e. Evenfin )

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2516  E.wrex 2615  (/)c0 3550  ~P₁ cpw1 4135   Nn cnnc 4373   +o cplc 4375   T_fin ctfin 4435   Even_fin cevenfin 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-tfin 4443  df-evenfin 4444

This theorem is referenced by:  vinf  4555