Theorem omcl 6350

Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)

Assertion

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

Proof of Theorem omcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omv 6344	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
2		0elon 4309	. . . 4 |- (/) e. On
3	2	a1i 9	. . 3 |- ( A e. On -> (/) e. On )
4		vex 2684	. . . . . . 7 |- y e. _V
5		oacl 6349	. . . . . . 7 |- ( ( y e. On /\ A e. On ) -> ( y +o A ) e. On )
6		oveq1 5774	. . . . . . . 8 |- ( x = y -> ( x +o A ) = ( y +o A ) )
7		eqid 2137	. . . . . . . 8 |- ( x e. _V |-> ( x +o A ) ) = ( x e. _V |-> ( x +o A ) )
8	6, 7	fvmptg 5490	. . . . . . 7 |- ( ( y e. _V /\ ( y +o A ) e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
9	4, 5, 8	sylancr 410	. . . . . 6 |- ( ( y e. On /\ A e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
10	9, 5	eqeltrd 2214	. . . . 5 |- ( ( y e. On /\ A e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) e. On )
11	10	ancoms 266	. . . 4 |- ( ( A e. On /\ y e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) e. On )
12	11	ralrimiva 2503	. . 3 |- ( A e. On -> A. y e. On ( ( x e. _V |-> ( x +o A ) ) ` y ) e. On )
13	3, 12	rdgon 6276	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. On )
14	1, 13	eqeltrd 2214	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2681   (/)c0 3358   |-> cmpt 3984   Oncon0 4280   ` cfv 5118  (class class class)co 5767   reccrdg 6259   +o coa 6303   .o comu 6304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311

This theorem is referenced by:  oeicl  6351  omv2  6354  omsuc  6361