Theorem oalim 4303

Description: Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56.

Assertion

Ref	Expression
oalim	|- ((A e. On /\ (B e. C /\ Lim B)) -> (A +o B) = U_x e. B (A +o x))

Distinct variable groups:   x,A   x,B

Proof of Theorem oalim

Step	Hyp	Ref	Expression
1		rdglim2a 4251	. . . 4 |- ((B e. On /\ Lim B) -> (rec({<.y, z>. | z = suc y}, A)` B) = U_x e. B (rec({<.y, z>. | z = suc y}, A)` x))
2	1	adantl 388	. . 3 |- ((A e. On /\ (B e. On /\ Lim B)) -> (rec({<.y, z>. | z = suc y}, A)` B) = U_x e. B (rec({<.y, z>. | z = suc y}, A)` x))
3		oav 4286	. . . . 5 |- ((A e. On /\ B e. On) -> (A +o B) = (rec({<.y, z>. | z = suc y}, A)` B))
4		oav 4286	. . . . . . . 8 |- ((A e. On /\ x e. On) -> (A +o x) = (rec({<.y, z>. | z = suc y}, A)` x))
5		onelon 3000	. . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6	4, 5	sylan2 453	. . . . . . 7 |- ((A e. On /\ (B e. On /\ x e. B)) -> (A +o x) = (rec({<.y, z>. | z = suc y}, A)` x))
7	6	anassrs 443	. . . . . 6 |- (((A e. On /\ B e. On) /\ x e. B) -> (A +o x) = (rec({<.y, z>. | z = suc y}, A)` x))
8	7	iuneq2dv 2650	. . . . 5 |- ((A e. On /\ B e. On) -> U_x e. B (A +o x) = U_x e. B (rec({<.y, z>. | z = suc y}, A)` x))
9	3, 8	eqeq12d 1532	. . . 4 |- ((A e. On /\ B e. On) -> ((A +o B) = U_x e. B (A +o x) <-> (rec({<.y, z>. | z = suc y}, A)` B) = U_x e. B (rec({<.y, z>. | z = suc y}, A)` x)))
10	9	adantrr 395	. . 3 |- ((A e. On /\ (B e. On /\ Lim B)) -> ((A +o B) = U_x e. B (A +o x) <-> (rec({<.y, z>. | z = suc y}, A)` B) = U_x e. B (rec({<.y, z>. | z = suc y}, A)` x)))
11	2, 10	mpbird 194	. 2 |- ((A e. On /\ (B e. On /\ Lim B)) -> (A +o B) = U_x e. B (A +o x))
12		limelon 3036	. . 3 |- ((B e. C /\ Lim B) -> B e. On)
13		pm3.27 321	. . 3 |- ((B e. C /\ Lim B) -> Lim B)
14	12, 13	jca 286	. 2 |- ((B e. C /\ Lim B) -> (B e. On /\ Lim B))
15	11, 14	sylan2 453	1 |- ((A e. On /\ (B e. C /\ Lim B)) -> (A +o B) = U_x e. B (A +o x))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  U_ciun 2633  {copab 2740  Oncon0 2975  Lim wlim 2976  suc csuc 2977  ` cfv 3263  (class class class)co 4021  reccrdg 4232   +o coa 4266

This theorem is referenced by:  oacl 4306  oa0r 4309  oaordi 4316  oawordri 4320  oawordeulem 4324  oalimcl 4330  oaass 4331  oarec 4332  odi 4346  oeoalem 4359  oaabslem 4391  oaabs 4392

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279  df-opr 4023  df-oprab 4024  df-rdg 4233  df-oadd 4271

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