Theorem oalim 4157

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 3941	. . . 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 4140	. . . . 5 |- ((A e. On /\ B e. On) -> (A +o B) = (rec({<.y, z>. | z = suc y}, A)` B))
4		oav 4140	. . . . . . . 8 |- ((A e. On /\ x e. On) -> (A +o x) = (rec({<.y, z>. | z = suc y}, A)` x))
5		onelon 2967	. . . . . . . 8 |- ((B e. On /\ x e. B) -> x e. On)
6	4, 5	sylan2 451	. . . . . . 7 |- ((A e. On /\ (B e. On /\ x e. B)) -> (A +o x) = (rec({<.y, z>. | z = suc y}, A)` x))
7	6	anassrs 441	. . . . . 6 |- (((A e. On /\ B e. On) /\ x e. B) -> (A +o x) = (rec({<.y, z>. | z = suc y}, A)` x))
8	7	iuneq2dv 2577	. . . . 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 1486	. . . 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 196	. 2 |- ((A e. On /\ (B e. On /\ Lim B)) -> (A +o B) = U_x e. B (A +o x))
12		limelon 3027	. . 3 |- ((B e. C /\ Lim B) -> B e. On)
13		pm3.27 323	. . 3 |- ((B e. C /\ Lim B) -> Lim B)
14	12, 13	jca 288	. 2 |- ((B e. C /\ Lim B) -> (B e. On /\ Lim B))
15	11, 14	sylan2 451	1 |- ((A e. On /\ (B e. C /\ Lim B)) -> (A +o B) = U_x e. B (A +o x))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  U_ciun 2561  {copab 2661  Oncon0 2943  Lim wlim 2944  suc csuc 2945  ` cfv 3177  reccrdg 3922  (class class class)co 3954   +o coa 4120

This theorem is referenced by:  oacl 4160  oa0r 4163  oaordi 4170  oawordri 4174  oawordeulem 4178  oalimcl 4184  oaass 4185  oarec 4186  odi 4200  oaabslem 4241  oaabs 4242

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-oadd 4125

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