Theorem oacl 5215

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 4890	. . . 4 |- (x = (/) -> (A +o x) = (A +o (/)))
2	1	eleq1d 1963	. . 3 |- (x = (/) -> ((A +o x) e. On <-> (A +o (/)) e. On))
3		opreq2 4890	. . . 4 |- (x = y -> (A +o x) = (A +o y))
4	3	eleq1d 1963	. . 3 |- (x = y -> ((A +o x) e. On <-> (A +o y) e. On))
5		opreq2 4890	. . . 4 |- (x = suc y -> (A +o x) = (A +o suc y))
6	5	eleq1d 1963	. . 3 |- (x = suc y -> ((A +o x) e. On <-> (A +o suc y) e. On))
7		opreq2 4890	. . . 4 |- (x = B -> (A +o x) = (A +o B))
8	7	eleq1d 1963	. . 3 |- (x = B -> ((A +o x) e. On <-> (A +o B) e. On))
9		oa0 5200	. . . . 5 |- (A e. On -> (A +o (/)) = A)
10	9	eleq1d 1963	. . . 4 |- (A e. On -> ((A +o (/)) e. On <-> A e. On))
11	10	ibir 653	. . 3 |- (A e. On -> (A +o (/)) e. On)
12		oasuc 5208	. . . . . 6 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
13	12	eleq1d 1963	. . . . 5 |- ((A e. On /\ y e. On) -> ((A +o suc y) e. On <-> suc (A +o y) e. On))
14		suceloni 3894	. . . . 5 |- ((A +o y) e. On -> suc (A +o y) e. On)
15	13, 14	syl5bir 227	. . . 4 |- ((A e. On /\ y e. On) -> ((A +o y) e. On -> (A +o suc y) e. On))
16	15	expcom 403	. . 3 |- (y e. On -> (A e. On -> ((A +o y) e. On -> (A +o suc y) e. On)))
17		visset 2295	. . . . . . 7 |- x e. _V
18		oalim 5212	. . . . . . 7 |- ((A e. On /\ (x e. _V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
19	17, 18	mpanr1 774	. . . . . 6 |- ((A e. On /\ Lim x) -> (A +o x) = U_y e. x (A +o y))
20	19	eleq1d 1963	. . . . 5 |- ((A e. On /\ Lim x) -> ((A +o x) e. On <-> U_y e. x (A +o y) e. On))
21		oprex 4907	. . . . . 6 |- (A +o y) e. _V
22	17, 21	iunon 5114	. . . . 5 |- (A.y e. x (A +o y) e. On -> U_y e. x (A +o y) e. On)
23	20, 22	syl5bir 227	. . . 4 |- ((A e. On /\ Lim x) -> (A.y e. x (A +o y) e. On -> (A +o x) e. On))
24	23	expcom 403	. . 3 |- (Lim x -> (A e. On -> (A.y e. x (A +o y) e. On -> (A +o x) e. On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 3948	. 2 |- (B e. On -> (A e. On -> (A +o B) e. On))
26	25	impcom 378	1 |- ((A e. On /\ B e. On) -> (A +o B) e. On)

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

This theorem is referenced by:  omcl 5216  omclOLD 5217  oaord 5228  oacan 5229  oaword 5230  oawordOLD 5231  oawordri 5232  oawordeulem 5236  oalimcl 5242  oaass 5243  odi 5258  oeoalem 5271  oeoa 5272  oancom 5740

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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