Theorem oa0r 5220

Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57.

Assertion

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

Proof of Theorem oa0r

Step	Hyp	Ref	Expression
1		opreq2 4890	. . 3 |- (x = (/) -> ((/) +o x) = ((/) +o (/)))
2		id 73	. . 3 |- (x = (/) -> x = (/))
3	1, 2	eqeq12d 1899	. 2 |- (x = (/) -> (((/) +o x) = x <-> ((/) +o (/)) = (/)))
4		opreq2 4890	. . 3 |- (x = y -> ((/) +o x) = ((/) +o y))
5		id 73	. . 3 |- (x = y -> x = y)
6	4, 5	eqeq12d 1899	. 2 |- (x = y -> (((/) +o x) = x <-> ((/) +o y) = y))
7		opreq2 4890	. . 3 |- (x = suc y -> ((/) +o x) = ((/) +o suc y))
8		id 73	. . 3 |- (x = suc y -> x = suc y)
9	7, 8	eqeq12d 1899	. 2 |- (x = suc y -> (((/) +o x) = x <-> ((/) +o suc y) = suc y))
10		opreq2 4890	. . 3 |- (x = A -> ((/) +o x) = ((/) +o A))
11		id 73	. . 3 |- (x = A -> x = A)
12	10, 11	eqeq12d 1899	. 2 |- (x = A -> (((/) +o x) = x <-> ((/) +o A) = A))
13		0elon 3716	. . 3 |- (/) e. On
14		oa0 5200	. . 3 |- ((/) e. On -> ((/) +o (/)) = (/))
15	13, 14	ax-mp 7	. 2 |- ((/) +o (/)) = (/)
16		oasuc 5208	. . . . 5 |- (((/) e. On /\ y e. On) -> ((/) +o suc y) = suc ((/) +o y))
17	13, 16	mpan 759	. . . 4 |- (y e. On -> ((/) +o suc y) = suc ((/) +o y))
18		suceq 3729	. . . 4 |- (((/) +o y) = y -> suc ((/) +o y) = suc y)
19	17, 18	sylan9eq 1948	. . 3 |- ((y e. On /\ ((/) +o y) = y) -> ((/) +o suc y) = suc y)
20	19	ex 402	. 2 |- (y e. On -> (((/) +o y) = y -> ((/) +o suc y) = suc y))
21		visset 2295	. . . . 5 |- x e. _V
22		oalim 5212	. . . . . 6 |- (((/) e. On /\ (x e. _V /\ Lim x)) -> ((/) +o x) = U_y e. x ((/) +o y))
23	13, 22	mpan 759	. . . . 5 |- ((x e. _V /\ Lim x) -> ((/) +o x) = U_y e. x ((/) +o y))
24	21, 23	mpan 759	. . . 4 |- (Lim x -> ((/) +o x) = U_y e. x ((/) +o y))
25		limuni 3724	. . . 4 |- (Lim x -> x = U.x)
26	24, 25	eqeq12d 1899	. . 3 |- (Lim x -> (((/) +o x) = x <-> U_y e. x ((/) +o y) = U.x))
27		iuneq2 3273	. . . 4 |- (A.y e. x ((/) +o y) = y -> U_y e. x ((/) +o y) = U_y e. x y)
28		uniiun 3306	. . . 4 |- U.x = U_y e. x y
29	27, 28	syl6eqr 1946	. . 3 |- (A.y e. x ((/) +o y) = y -> U_y e. x ((/) +o y) = U.x)
30	26, 29	syl5bir 227	. 2 |- (Lim x -> (A.y e. x ((/) +o y) = y -> ((/) +o x) = x))
31	3, 6, 9, 12, 15, 20, 30	tfinds 3942	1 |- (A e. On -> ((/) +o A) = A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  (/)c0 2875  U.cuni 3177  U_ciun 3255  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884   +o coa 5174

This theorem is referenced by:  om1 5223  oaword2 5235  nna0r 5279

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-oadd 5179

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