Theorem oa0r 4157

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 3954	. . 3 |- (x = (/) -> ((/) +o x) = ((/) +o (/)))
2		id 59	. . 3 |- (x = (/) -> x = (/))
3	1, 2	eqeq12d 1481	. 2 |- (x = (/) -> (((/) +o x) = x <-> ((/) +o (/)) = (/)))
4		opreq2 3954	. . 3 |- (x = y -> ((/) +o x) = ((/) +o y))
5		id 59	. . 3 |- (x = y -> x = y)
6	4, 5	eqeq12d 1481	. 2 |- (x = y -> (((/) +o x) = x <-> ((/) +o y) = y))
7		opreq2 3954	. . 3 |- (x = suc y -> ((/) +o x) = ((/) +o suc y))
8		id 59	. . 3 |- (x = suc y -> x = suc y)
9	7, 8	eqeq12d 1481	. 2 |- (x = suc y -> (((/) +o x) = x <-> ((/) +o suc y) = suc y))
10		opreq2 3954	. . 3 |- (x = A -> ((/) +o x) = ((/) +o A))
11		id 59	. . 3 |- (x = A -> x = A)
12	10, 11	eqeq12d 1481	. 2 |- (x = A -> (((/) +o x) = x <-> ((/) +o A) = A))
13		0elon 3012	. . 3 |- (/) e. On
14		oa0 4139	. . 3 |- ((/) e. On -> ((/) +o (/)) = (/))
15	13, 14	ax-mp 7	. 2 |- ((/) +o (/)) = (/)
16		oasuc 4147	. . . . 5 |- (((/) e. On /\ y e. On) -> ((/) +o suc y) = suc ((/) +o y))
17	13, 16	mpan 693	. . . 4 |- (y e. On -> ((/) +o suc y) = suc ((/) +o y))
18		suceq 3024	. . . 4 |- (((/) +o y) = y -> suc ((/) +o y) = suc y)
19	17, 18	sylan9eq 1519	. . 3 |- ((y e. On /\ ((/) +o y) = y) -> ((/) +o suc y) = suc y)
20	19	ex 373	. 2 |- (y e. On -> (((/) +o y) = y -> ((/) +o suc y) = suc y))
21		visset 1804	. . . . 5 |- x e. V
22		oalim 4151	. . . . . 6 |- (((/) e. On /\ (x e. V /\ Lim x)) -> ((/) +o x) = U_y e. x ((/) +o y))
23	13, 22	mpan 693	. . . . 5 |- ((x e. V /\ Lim x) -> ((/) +o x) = U_y e. x ((/) +o y))
24	21, 23	mpan 693	. . . 4 |- (Lim x -> ((/) +o x) = U_y e. x ((/) +o y))
25		limuni 3019	. . . 4 |- (Lim x -> x = U.x)
26	24, 25	eqeq12d 1481	. . 3 |- (Lim x -> (((/) +o x) = x <-> U_y e. x ((/) +o y) = U.x))
27		iuneq2 2568	. . . 4 |- (A.y e. x ((/) +o y) = y -> U_y e. x ((/) +o y) = U_y e. x y)
28		uniiun 2591	. . . 4 |- U.x = U_y e. x y
29	27, 28	syl6eqr 1517	. . 3 |- (A.y e. x ((/) +o y) = y -> U_y e. x ((/) +o y) = U.x)
30	26, 29	syl5bir 210	. 2 |- (Lim x -> (A.y e. x ((/) +o y) = y -> ((/) +o x) = x))
31	3, 6, 9, 12, 15, 20, 30	tfinds 3151	1 |- (A e. On -> ((/) +o A) = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802  (/)c0 2270  U.cuni 2493  U_ciun 2556  Oncon0 2938  Lim wlim 2939  suc csuc 2940  (class class class)co 3948   +o coa 4114

This theorem is referenced by:  om1 4160  oaword2 4171  nna0r 4211

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-oadd 4119

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