Theorem oacl 4170

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 3969	. . . 4 |- (x = (/) -> (A +o x) = (A +o (/)))
2	1	eleq1d 1540	. . 3 |- (x = (/) -> ((A +o x) e. On <-> (A +o (/)) e. On))
3		opreq2 3969	. . . 4 |- (x = y -> (A +o x) = (A +o y))
4	3	eleq1d 1540	. . 3 |- (x = y -> ((A +o x) e. On <-> (A +o y) e. On))
5		opreq2 3969	. . . 4 |- (x = suc y -> (A +o x) = (A +o suc y))
6	5	eleq1d 1540	. . 3 |- (x = suc y -> ((A +o x) e. On <-> (A +o suc y) e. On))
7		opreq2 3969	. . . 4 |- (x = B -> (A +o x) = (A +o B))
8	7	eleq1d 1540	. . 3 |- (x = B -> ((A +o x) e. On <-> (A +o B) e. On))
9		oa0 4155	. . . . 5 |- (A e. On -> (A +o (/)) = A)
10	9	eleq1d 1540	. . . 4 |- (A e. On -> ((A +o (/)) e. On <-> A e. On))
11	10	ibir 593	. . 3 |- (A e. On -> (A +o (/)) e. On)
12		oasuc 4163	. . . . . 6 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
13	12	eleq1d 1540	. . . . 5 |- ((A e. On /\ y e. On) -> ((A +o suc y) e. On <-> suc (A +o y) e. On))
14		suceloni 3062	. . . . 5 |- ((A +o y) e. On -> suc (A +o y) e. On)
15	13, 14	syl5bir 210	. . . 4 |- ((A e. On /\ y e. On) -> ((A +o y) e. On -> (A +o suc y) e. On))
16	15	expcom 374	. . 3 |- (y e. On -> (A e. On -> ((A +o y) e. On -> (A +o suc y) e. On)))
17		visset 1813	. . . . . . 7 |- x e. V
18		oalim 4167	. . . . . . 7 |- ((A e. On /\ (x e. V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
19	17, 18	mpanr1 709	. . . . . 6 |- ((A e. On /\ Lim x) -> (A +o x) = U_y e. x (A +o y))
20	19	eleq1d 1540	. . . . 5 |- ((A e. On /\ Lim x) -> ((A +o x) e. On <-> U_y e. x (A +o y) e. On))
21		oprex 3983	. . . . . 6 |- (A +o y) e. V
22	17, 21	iunon 3909	. . . . 5 |- (A.y e. x (A +o y) e. On -> U_y e. x (A +o y) e. On)
23	20, 22	syl5bir 210	. . . 4 |- ((A e. On /\ Lim x) -> (A.y e. x (A +o y) e. On -> (A +o x) e. On))
24	23	expcom 374	. . 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 3166	. 2 |- (B e. On -> (A e. On -> (A +o B) e. On))
26	25	impcom 351	1 |- ((A e. On /\ B e. On) -> (A +o B) e. On)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  (/)c0 2280  U_ciun 2566  Oncon0 2948  Lim wlim 2949  suc csuc 2950  (class class class)co 3963   +o coa 4130

This theorem is referenced by:  omcl 4171  oaord 4181  oacan 4182  oaword 4183  oawordri 4184  oawordeulem 4188  oalimcl 4194  oaass 4195  odi 4210  oancom 4633

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-oadd 4135

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