Theorem omlim 4161

Description: Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62.

Assertion

Ref	Expression
omlim	|- ((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 omlim

Step	Hyp	Ref	Expression
1		rdglim2a 3945	. . . 4 |- ((B e. On /\ Lim B) -> (rec({<.y, z>. | z = (y +o A)}, (/))` B) = U_x e. B (rec({<.y, z>. | z = (y +o A)}, (/))` x))
2	1	adantl 388	. . 3 |- ((A e. On /\ (B e. On /\ Lim B)) -> (rec({<.y, z>. | z = (y +o A)}, (/))` B) = U_x e. B (rec({<.y, z>. | z = (y +o A)}, (/))` x))
3		omv 4144	. . . . 5 |- ((A e. On /\ B e. On) -> (A .o B) = (rec({<.y, z>. | z = (y +o A)}, (/))` B))
4		omv 4144	. . . . . . . 8 |- ((A e. On /\ x e. On) -> (A .o x) = (rec({<.y, z>. | z = (y +o A)}, (/))` x))
5		onelon 2968	. . . . . . . 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 = (y +o A)}, (/))` x))
7	6	anassrs 441	. . . . . 6 |- (((A e. On /\ B e. On) /\ x e. B) -> (A .o x) = (rec({<.y, z>. | z = (y +o A)}, (/))` x))
8	7	iuneq2dv 2578	. . . . 5 |- ((A e. On /\ B e. On) -> U_x e. B (A .o x) = U_x e. B (rec({<.y, z>. | z = (y +o A)}, (/))` x))
9	3, 8	eqeq12d 1487	. . . 4 |- ((A e. On /\ B e. On) -> ((A .o B) = U_x e. B (A .o x) <-> (rec({<.y, z>. | z = (y +o A)}, (/))` B) = U_x e. B (rec({<.y, z>. | z = (y +o 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 = (y +o A)}, (/))` B) = U_x e. B (rec({<.y, z>. | z = (y +o 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 3028	. . 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 955   e. wcel 957  (/)c0 2277  U_ciun 2562  {copab 2662  Oncon0 2944  Lim wlim 2945  ` cfv 3178  reccrdg 3926  (class class class)co 3958   +o coa 4123   .o comu 4124

This theorem is referenced by:  omcl 4164  om0r 4167  om1r 4170  omordi 4190  omwordri 4196  omordlim 4201  omlimcl 4202  odi 4203  omass 4204

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-omul 4129

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