Theorem oa0r 6976

Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Assertion

Ref	Expression
oa0r	|- ( A e. On -> ( (/) +o A ) = A )

Proof of Theorem oa0r

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6097	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 22	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2455	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 6097	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 22	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2455	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 6097	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 22	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2455	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 6097	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 22	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2455	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4770	. . 3 |- (/) e. On
14		oa0 6954	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		oasuc 6962	. . . . 5 |- ( ( (/) e. On /\ y e. On ) -> ( (/) +o suc y ) = suc ( (/) +o y ) )
17	13, 16	mpan 670	. . . 4 |- ( y e. On -> ( (/) +o suc y ) = suc ( (/) +o y ) )
18		suceq 4782	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
19	17, 18	sylan9eq 2493	. . 3 |- ( ( y e. On /\ ( (/) +o y ) = y ) -> ( (/) +o suc y ) = suc y )
20	19	ex 434	. 2 |- ( y e. On -> ( ( (/) +o y ) = y -> ( (/) +o suc y ) = suc y ) )
21		iuneq2 4185	. . . 4 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U_ y e. x y )
22		uniiun 4221	. . . 4 |- U. x = U_ y e. x y
23	21, 22	syl6eqr 2491	. . 3 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U. x )
24		vex 2973	. . . . 5 |- x e. _V
25		oalim 6970	. . . . . 6 |- ( ( (/) e. On /\ ( x e. _V /\ Lim x ) ) -> ( (/) +o x ) = U_ y e. x ( (/) +o y ) )
26	13, 25	mpan 670	. . . . 5 |- ( ( x e. _V /\ Lim x ) -> ( (/) +o x ) = U_ y e. x ( (/) +o y ) )
27	24, 26	mpan 670	. . . 4 |- ( Lim x -> ( (/) +o x ) = U_ y e. x ( (/) +o y ) )
28		limuni 4777	. . . 4 |- ( Lim x -> x = U. x )
29	27, 28	eqeq12d 2455	. . 3 |- ( Lim x -> ( ( (/) +o x ) = x <-> U_ y e. x ( (/) +o y ) = U. x ) )
30	23, 29	syl5ibr 221	. 2 |- ( Lim x -> ( A. y e. x ( (/) +o y ) = y -> ( (/) +o x ) = x ) )
31	3, 6, 9, 12, 15, 20, 30	tfinds 6468	1 |- ( A e. On -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2713   _Vcvv 2970   (/)c0 3635   U.cuni 4089   U_ciun 4169   Oncon0 4717   Lim wlim 4718   suc csuc 4719  (class class class)co 6089   +o coa 6915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-oadd 6922

This theorem is referenced by:  om1  6979  oaword2  6990  oeeui  7039  oaabs2  7082  cantnfp1  7887  cantnfp1OLD  7913