Theorem oa0r 7206

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 6304	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 22	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2479	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 6304	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 22	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2479	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 6304	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 22	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2479	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 6304	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 22	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2479	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4940	. . 3 |- (/) e. On
14		oa0 7184	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		oasuc 7192	. . . . 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 4952	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
19	17, 18	sylan9eq 2518	. . 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 4349	. . . 4 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U_ y e. x y )
22		uniiun 4385	. . . 4 |- U. x = U_ y e. x y
23	21, 22	syl6eqr 2516	. . 3 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U. x )
24		vex 3112	. . . . 5 |- x e. _V
25		oalim 7200	. . . . . 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 4947	. . . 4 |- ( Lim x -> x = U. x )
29	27, 28	eqeq12d 2479	. . 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 6693	1 |- ( A e. On -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   _Vcvv 3109   (/)c0 3793   U.cuni 4251   U_ciun 4332   Oncon0 4887   Lim wlim 4888   suc csuc 4889  (class class class)co 6296   +o coa 7145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-oadd 7152

This theorem is referenced by:  om1  7209  oaword2  7220  oeeui  7269  oaabs2  7312  cantnfp1  8117  cantnfp1OLD  8143