Theorem oacl 3138

Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57.

Assertion

Ref	Expression
oacl	|- ((A e. On /\ B e. On) -> (A +o B) e. On)

Proof of Theorem oacl

Step	Hyp	Ref	Expression
1		opreq2 3007	. . . . 5 |- (x = (/) -> (A +o x) = (A +o (/)))
2	1	eleq1d 1155	. . . 4 |- (x = (/) -> ((A +o x) e. On <-> (A +o (/)) e. On))
3		opreq2 3007	. . . . 5 |- (x = y -> (A +o x) = (A +o y))
4	3	eleq1d 1155	. . . 4 |- (x = y -> ((A +o x) e. On <-> (A +o y) e. On))
5		opreq2 3007	. . . . 5 |- (x = suc y -> (A +o x) = (A +o suc y))
6	5	eleq1d 1155	. . . 4 |- (x = suc y -> ((A +o x) e. On <-> (A +o suc y) e. On))
7		opreq2 3007	. . . . 5 |- (x = B -> (A +o x) = (A +o B))
8	7	eleq1d 1155	. . . 4 |- (x = B -> ((A +o x) e. On <-> (A +o B) e. On))
9		oa0 3124	. . . . . 6 |- (A e. On -> (A +o (/)) = A)
10	9	eleq1d 1155	. . . . 5 |- (A e. On -> ((A +o (/)) e. On <-> A e. On))
11	10	ibir 450	. . . 4 |- (A e. On -> (A +o (/)) e. On)
12		oasuc 3131	. . . . . . . 8 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
13	12	eleq1d 1155	. . . . . . 7 |- ((A e. On /\ y e. On) -> ((A +o suc y) e. On <-> suc (A +o y) e. On))
14		suceloni 2314	. . . . . . 7 |- ((A +o y) e. On -> suc (A +o y) e. On)
15	13, 14	syl5bir 184	. . . . . 6 |- ((A e. On /\ y e. On) -> ((A +o y) e. On -> (A +o suc y) e. On))
16	15	exp 291	. . . . 5 |- (A e. On -> (y e. On -> ((A +o y) e. On -> (A +o suc y) e. On)))
17	16	com12 13	. . . 4 |- (y e. On -> (A e. On -> ((A +o y) e. On -> (A +o suc y) e. On)))
18		visset 1350	. . . . . . . . 9 |- x e. V
19		oalim 3135	. . . . . . . . 9 |- ((A e. On /\ (x e. V /\ Lim x)) -> (A +o x) = U.y e. x (A +o y))
20	18, 19	mpan21 531	. . . . . . . 8 |- ((A e. On /\ Lim x) -> (A +o x) = U.y e. x (A +o y))
21	20	eleq1d 1155	. . . . . . 7 |- ((A e. On /\ Lim x) -> ((A +o x) e. On <-> U.y e. x (A +o y) e. On))
22		oprex 3018	. . . . . . . 8 |- (A +o y) e. V
23	18, 22	iunon 2947	. . . . . . 7 |- (A.y e. x (A +o y) e. On -> U.y e. x (A +o y) e. On)
24	21, 23	syl5bir 184	. . . . . 6 |- ((A e. On /\ Lim x) -> (A.y e. x (A +o y) e. On -> (A +o x) e. On))
25	24	exp 291	. . . . 5 |- (A e. On -> (Lim x -> (A.y e. x (A +o y) e. On -> (A +o x) e. On)))
26	25	com12 13	. . . 4 |- (Lim x -> (A e. On -> (A.y e. x (A +o y) e. On -> (A +o x) e. On)))
27	2, 4, 6, 8, 11, 17, 26	tfinds3 2406	. . 3 |- (B e. On -> (A e. On -> (A +o B) e. On))
28	27	com12 13	. 2 |- (A e. On -> (B e. On -> (A +o B) e. On))
29	28	imp 277	1 |- ((A e. On /\ B e. On) -> (A +o B) e. On)

Syntax hints:   -> wi 2   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348  (/)c0 1707  U.ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001   +o coa 3101

This theorem is referenced by:  omcl 3139  oaord 3149  oacan 3150  oaword 3151  oawordri 3152  oawordeulem 3156  oalimcl 3162  oaass 3163

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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