Theorem oacl 4176

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 3975	. . . 4 |- (x = (/) -> (A +o x) = (A +o (/)))
2	1	eleq1d 1543	. . 3 |- (x = (/) -> ((A +o x) e. On <-> (A +o (/)) e. On))
3		opreq2 3975	. . . 4 |- (x = y -> (A +o x) = (A +o y))
4	3	eleq1d 1543	. . 3 |- (x = y -> ((A +o x) e. On <-> (A +o y) e. On))
5		opreq2 3975	. . . 4 |- (x = suc y -> (A +o x) = (A +o suc y))
6	5	eleq1d 1543	. . 3 |- (x = suc y -> ((A +o x) e. On <-> (A +o suc y) e. On))
7		opreq2 3975	. . . 4 |- (x = B -> (A +o x) = (A +o B))
8	7	eleq1d 1543	. . 3 |- (x = B -> ((A +o x) e. On <-> (A +o B) e. On))
9		oa0 4161	. . . . 5 |- (A e. On -> (A +o (/)) = A)
10	9	eleq1d 1543	. . . 4 |- (A e. On -> ((A +o (/)) e. On <-> A e. On))
11	10	ibir 595	. . 3 |- (A e. On -> (A +o (/)) e. On)
12		oasuc 4169	. . . . . 6 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
13	12	eleq1d 1543	. . . . 5 |- ((A e. On /\ y e. On) -> ((A +o suc y) e. On <-> suc (A +o y) e. On))
14		suceloni 3068	. . . . 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 1816	. . . . . . 7 |- x e. V
18		oalim 4173	. . . . . . 7 |- ((A e. On /\ (x e. V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
19	17, 18	mpanr1 711	. . . . . 6 |- ((A e. On /\ Lim x) -> (A +o x) = U_y e. x (A +o y))
20	19	eleq1d 1543	. . . . 5 |- ((A e. On /\ Lim x) -> ((A +o x) e. On <-> U_y e. x (A +o y) e. On))
21		oprex 3989	. . . . . 6 |- (A +o y) e. V
22	17, 21	iunon 3915	. . . . 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 3172	. 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 958   e. wcel 960  A.wral 1648  Vcvv 1814  (/)c0 2283  U_ciun 2570  Oncon0 2954  Lim wlim 2955  suc csuc 2956  (class class class)co 3969   +o coa 4136

This theorem is referenced by:  omcl 4177  oaord 4187  oacan 4188  oaword 4189  oawordri 4190  oawordeulem 4194  oalimcl 4200  oaass 4201  odi 4216  oancom 4642

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-oadd 4141

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