Theorem omcl 6311

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 6305	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
2		0elon 4274	. . . 4 |- (/) e. On
3	2	a1i 9	. . 3 |- ( A e. On -> (/) e. On )
4		vex 2660	. . . . . . 7 |- y e. _V
5		oacl 6310	. . . . . . 7 |- ( ( y e. On /\ A e. On ) -> ( y +o A ) e. On )
6		oveq1 5735	. . . . . . . 8 |- ( x = y -> ( x +o A ) = ( y +o A ) )
7		eqid 2115	. . . . . . . 8 |- ( x e. _V |-> ( x +o A ) ) = ( x e. _V |-> ( x +o A ) )
8	6, 7	fvmptg 5451	. . . . . . 7 |- ( ( y e. _V /\ ( y +o A ) e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
9	4, 5, 8	sylancr 408	. . . . . 6 |- ( ( y e. On /\ A e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
10	9, 5	eqeltrd 2191	. . . . 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 2479	. . 3 |- ( A e. On -> A. y e. On ( ( x e. _V |-> ( x +o A ) ) ` y ) e. On )
13	3, 12	rdgon 6237	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. On )
14	1, 13	eqeltrd 2191	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1314   e. wcel 1463   _Vcvv 2657   (/)c0 3329   |-> cmpt 3949   Oncon0 4245   ` cfv 5081  (class class class)co 5728   reccrdg 6220   +o coa 6264   .o comu 6265

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-oadd 6271  df-omul 6272

This theorem is referenced by:  oeicl  6312  omv2  6315  omsuc  6322