Theorem omv 6481

Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
omv	|- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem omv

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

Step	Hyp	Ref	Expression
1		0elon 4410	. . 3 |- (/) e. On
2		omfnex 6475	. . . 4 |- ( A e. On -> ( x e. _V |-> ( x +o A ) ) Fn _V )
3		rdgexggg 6403	. . . 4 |- ( ( ( x e. _V |-> ( x +o A ) ) Fn _V /\ (/) e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V )
4	2, 3	syl3an1 1282	. . 3 |- ( ( A e. On /\ (/) e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V )
5	1, 4	mp3an2 1336	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V )
6		oveq2 5905	. . . . . 6 |- ( y = A -> ( x +o y ) = ( x +o A ) )
7	6	mpteq2dv 4109	. . . . 5 |- ( y = A -> ( x e. _V |-> ( x +o y ) ) = ( x e. _V |-> ( x +o A ) ) )
8		rdgeq1 6397	. . . . 5 |- ( ( x e. _V |-> ( x +o y ) ) = ( x e. _V |-> ( x +o A ) ) -> rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) = rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) )
9	7, 8	syl 14	. . . 4 |- ( y = A -> rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) = rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) )
10	9	fveq1d 5536	. . 3 |- ( y = A -> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) )
11		fveq2 5534	. . 3 |- ( z = B -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
12		df-omul 6447	. . 3 |- .o = ( y e. On , z e. On |-> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) )
13	10, 11, 12	ovmpog 6032	. 2 |- ( ( A e. On /\ B e. On /\ ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
14	5, 13	mpd3an3 1349	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   (/)c0 3437   |-> cmpt 4079   Oncon0 4381   Fn wfn 5230   ` cfv 5235  (class class class)co 5897   reccrdg 6395   +o coa 6439   .o comu 6440

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447

This theorem is referenced by:  om0  6484  omcl  6487  omv2  6491