Theorem omcl 6429

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 6423	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
2		0elon 4370	. . . 4 |- (/) e. On
3	2	a1i 9	. . 3 |- ( A e. On -> (/) e. On )
4		vex 2729	. . . . . . 7 |- y e. _V
5		oacl 6428	. . . . . . 7 |- ( ( y e. On /\ A e. On ) -> ( y +o A ) e. On )
6		oveq1 5849	. . . . . . . 8 |- ( x = y -> ( x +o A ) = ( y +o A ) )
7		eqid 2165	. . . . . . . 8 |- ( x e. _V |-> ( x +o A ) ) = ( x e. _V |-> ( x +o A ) )
8	6, 7	fvmptg 5562	. . . . . . 7 |- ( ( y e. _V /\ ( y +o A ) e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
9	4, 5, 8	sylancr 411	. . . . . 6 |- ( ( y e. On /\ A e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
10	9, 5	eqeltrd 2243	. . . . 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 2539	. . 3 |- ( A e. On -> A. y e. On ( ( x e. _V |-> ( x +o A ) ) ` y ) e. On )
13	3, 12	rdgon 6354	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. On )
14	1, 13	eqeltrd 2243	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   (/)c0 3409   |-> cmpt 4043   Oncon0 4341   ` cfv 5188  (class class class)co 5842   reccrdg 6337   +o coa 6381   .o comu 6382

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389

This theorem is referenced by:  oeicl  6430  omv2  6433  omsuc  6440