Theorem oacl 6627

Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)

Assertion

Ref	Expression
oacl	|- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On )

Proof of Theorem oacl

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oav 6621	. 2 |- ( ( A e. On /\ B e. On ) -> ( A +o B ) = ( rec ( ( z e. _V |-> suc z ) , A ) ` B ) )
2		id 19	. . 3 |- ( A e. On -> A e. On )
3		vex 2805	. . . . . . . 8 |- w e. _V
4		suceq 4499	. . . . . . . . 9 |- ( z = w -> suc z = suc w )
5		eqid 2231	. . . . . . . . 9 |- ( z e. _V |-> suc z ) = ( z e. _V |-> suc z )
6	3	sucex 4597	. . . . . . . . 9 |- suc w e. _V
7	4, 5, 6	fvmpt 5723	. . . . . . . 8 |- ( w e. _V -> ( ( z e. _V |-> suc z ) ` w ) = suc w )
8	3, 7	ax-mp 5	. . . . . . 7 |- ( ( z e. _V |-> suc z ) ` w ) = suc w
9	8	eleq1i 2297	. . . . . 6 |- ( ( ( z e. _V |-> suc z ) ` w ) e. On <-> suc w e. On )
10	9	ralbii 2538	. . . . 5 |- ( A. w e. On ( ( z e. _V |-> suc z ) ` w ) e. On <-> A. w e. On suc w e. On )
11		onsuc 4599	. . . . 5 |- ( w e. On -> suc w e. On )
12	10, 11	mprgbir 2590	. . . 4 |- A. w e. On ( ( z e. _V |-> suc z ) ` w ) e. On
13	12	a1i 9	. . 3 |- ( A e. On -> A. w e. On ( ( z e. _V |-> suc z ) ` w ) e. On )
14	2, 13	rdgon 6551	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( z e. _V |-> suc z ) , A ) ` B ) e. On )
15	1, 14	eqeltrd 2308	1 |- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   |-> cmpt 4150   Oncon0 4460   suc csuc 4462   ` cfv 5326  (class class class)co 6017   reccrdg 6534   +o coa 6578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-irdg 6535  df-oadd 6585

This theorem is referenced by:  omcl  6628  omv2  6632