Theorem oa0r 4173

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 3969	. . 3 |- (x = (/) -> ((/) +o x) = ((/) +o (/)))
2		id 59	. . 3 |- (x = (/) -> x = (/))
3	1, 2	eqeq12d 1489	. 2 |- (x = (/) -> (((/) +o x) = x <-> ((/) +o (/)) = (/)))
4		opreq2 3969	. . 3 |- (x = y -> ((/) +o x) = ((/) +o y))
5		id 59	. . 3 |- (x = y -> x = y)
6	4, 5	eqeq12d 1489	. 2 |- (x = y -> (((/) +o x) = x <-> ((/) +o y) = y))
7		opreq2 3969	. . 3 |- (x = suc y -> ((/) +o x) = ((/) +o suc y))
8		id 59	. . 3 |- (x = suc y -> x = suc y)
9	7, 8	eqeq12d 1489	. 2 |- (x = suc y -> (((/) +o x) = x <-> ((/) +o suc y) = suc y))
10		opreq2 3969	. . 3 |- (x = A -> ((/) +o x) = ((/) +o A))
11		id 59	. . 3 |- (x = A -> x = A)
12	10, 11	eqeq12d 1489	. 2 |- (x = A -> (((/) +o x) = x <-> ((/) +o A) = A))
13		0elon 3022	. . 3 |- (/) e. On
14		oa0 4155	. . 3 |- ((/) e. On -> ((/) +o (/)) = (/))
15	13, 14	ax-mp 7	. 2 |- ((/) +o (/)) = (/)
16		oasuc 4163	. . . . 5 |- (((/) e. On /\ y e. On) -> ((/) +o suc y) = suc ((/) +o y))
17	13, 16	mpan 695	. . . 4 |- (y e. On -> ((/) +o suc y) = suc ((/) +o y))
18		suceq 3034	. . . 4 |- (((/) +o y) = y -> suc ((/) +o y) = suc y)
19	17, 18	sylan9eq 1527	. . 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 1813	. . . . 5 |- x e. V
22		oalim 4167	. . . . . 6 |- (((/) e. On /\ (x e. V /\ Lim x)) -> ((/) +o x) = U_y e. x ((/) +o y))
23	13, 22	mpan 695	. . . . 5 |- ((x e. V /\ Lim x) -> ((/) +o x) = U_y e. x ((/) +o y))
24	21, 23	mpan 695	. . . 4 |- (Lim x -> ((/) +o x) = U_y e. x ((/) +o y))
25		limuni 3029	. . . 4 |- (Lim x -> x = U.x)
26	24, 25	eqeq12d 1489	. . 3 |- (Lim x -> (((/) +o x) = x <-> U_y e. x ((/) +o y) = U.x))
27		iuneq2 2578	. . . 4 |- (A.y e. x ((/) +o y) = y -> U_y e. x ((/) +o y) = U_y e. x y)
28		uniiun 2601	. . . 4 |- U.x = U_y e. x y
29	27, 28	syl6eqr 1525	. . 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 3161	1 |- (A e. On -> ((/) +o A) = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  (/)c0 2280  U.cuni 2503  U_ciun 2566  Oncon0 2948  Lim wlim 2949  suc csuc 2950  (class class class)co 3963   +o coa 4130

This theorem is referenced by:  om1 4176  oaword2 4187  nna0r 4227

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-oadd 4135

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