Theorem omcl 6440

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 6434	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
2		0elon 4377	. . . 4 |- (/) e. On
3	2	a1i 9	. . 3 |- ( A e. On -> (/) e. On )
4		vex 2733	. . . . . . 7 |- y e. _V
5		oacl 6439	. . . . . . 7 |- ( ( y e. On /\ A e. On ) -> ( y +o A ) e. On )
6		oveq1 5860	. . . . . . . 8 |- ( x = y -> ( x +o A ) = ( y +o A ) )
7		eqid 2170	. . . . . . . 8 |- ( x e. _V |-> ( x +o A ) ) = ( x e. _V |-> ( x +o A ) )
8	6, 7	fvmptg 5572	. . . . . . 7 |- ( ( y e. _V /\ ( y +o A ) e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
9	4, 5, 8	sylancr 412	. . . . . 6 |- ( ( y e. On /\ A e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` y ) = ( y +o A ) )
10	9, 5	eqeltrd 2247	. . . . 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 2543	. . 3 |- ( A e. On -> A. y e. On ( ( x e. _V |-> ( x +o A ) ) ` y ) e. On )
13	3, 12	rdgon 6365	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. On )
14	1, 13	eqeltrd 2247	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   (/)c0 3414   |-> cmpt 4050   Oncon0 4348   ` cfv 5198  (class class class)co 5853   reccrdg 6348   +o coa 6392   .o comu 6393

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400

This theorem is referenced by:  oeicl  6441  omv2  6444  omsuc  6451