Theorem oa0r 7178

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 6283	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 22	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2482	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 6283	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 22	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2482	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 6283	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 22	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2482	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 6283	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 22	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2482	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4924	. . 3 |- (/) e. On
14		oa0 7156	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		oasuc 7164	. . . . 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 4936	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
19	17, 18	sylan9eq 2521	. . 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 4335	. . . 4 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U_ y e. x y )
22		uniiun 4371	. . . 4 |- U. x = U_ y e. x y
23	21, 22	syl6eqr 2519	. . 3 |- ( A. y e. x ( (/) +o y ) = y -> U_ y e. x ( (/) +o y ) = U. x )
24		vex 3109	. . . . 5 |- x e. _V
25		oalim 7172	. . . . . 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 4931	. . . 4 |- ( Lim x -> x = U. x )
29	27, 28	eqeq12d 2482	. . 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 6665	1 |- ( A e. On -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   A.wral 2807   _Vcvv 3106   (/)c0 3778   U.cuni 4238   U_ciun 4318   Oncon0 4871   Lim wlim 4872   suc csuc 4873  (class class class)co 6275   +o coa 7117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-oadd 7124

This theorem is referenced by:  om1  7181  oaword2  7192  oeeui  7241  oaabs2  7284  cantnfp1  8089  cantnfp1OLD  8115