Theorem oa0r 7245

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 6303	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 22	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2468	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 6303	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 22	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2468	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 6303	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 22	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2468	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 6303	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 22	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2468	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 5479	. . 3 |- (/) e. On
14		oa0 7223	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		oasuc 7231	. . . . 5 |- ( ( (/) e. On /\ y e. On ) -> ( (/) +o suc y ) = suc ( (/) +o y ) )
17	13, 16	mpan 677	. . . 4 |- ( y e. On -> ( (/) +o suc y ) = suc ( (/) +o y ) )
18		suceq 5491	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
19	17, 18	sylan9eq 2507	. . 3 |- ( ( y e. On /\ ( (/) +o y ) = y ) -> ( (/) +o suc y ) = suc y )
20	19	ex 436	. 2 |- ( y e. On -> ( ( (/) +o y ) = y -> ( (/) +o suc y ) = suc y ) )
21		iuneq2 4298	. . . 4 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U_ y e. x y )
22		uniiun 4334	. . . 4 |- U. x = U_ y e. x y
23	21, 22	syl6eqr 2505	. . 3 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U. x )
24		vex 3050	. . . . 5 |- x e. _V
25		oalim 7239	. . . . . 6 |- ( ( (/) e. On /\ ( x e. _V /\ Lim x ) ) -> ( (/) +o x ) = U_ y e. x ( (/) +o y ) )
26	13, 25	mpan 677	. . . . 5 |- ( ( x e. _V /\ Lim x ) -> ( (/) +o x ) = U_ y e. x ( (/) +o y ) )
27	24, 26	mpan 677	. . . 4 |- ( Lim x -> ( (/) +o x ) = U_ y e. x ( (/) +o y ) )
28		limuni 5486	. . . 4 |- ( Lim x -> x = U. x )
29	27, 28	eqeq12d 2468	. . 3 |- ( Lim x -> ( ( (/) +o x ) = x <-> U_ y e. x ( (/) +o y ) = U. x ) )
30	23, 29	syl5ibr 225	. 2 |- ( Lim x -> ( A. y e. x ( (/) +o y ) = y -> ( (/) +o x ) = x ) )
31	3, 6, 9, 12, 15, 20, 30	tfinds 6691	1 |- ( A e. On -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   _Vcvv 3047   (/)c0 3733   U.cuni 4201   U_ciun 4281   Oncon0 5426   Lim wlim 5427   suc csuc 5428  (class class class)co 6295   +o coa 7184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-oadd 7191

This theorem is referenced by:  om1  7248  oaword2  7259  oeeui  7308  oaabs2  7351  cantnfp1  8191