Theorem omv 6510

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 4424	. . 3 |- (/) e. On
2		omfnex 6504	. . . 4 |- ( A e. On -> ( x e. _V |-> ( x +o A ) ) Fn _V )
3		rdgexggg 6432	. . . 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 5927	. . . . . 6 |- ( y = A -> ( x +o y ) = ( x +o A ) )
7	6	mpteq2dv 4121	. . . . 5 |- ( y = A -> ( x e. _V |-> ( x +o y ) ) = ( x e. _V |-> ( x +o A ) ) )
8		rdgeq1 6426	. . . . 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 5557	. . 3 |- ( y = A -> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) )
11		fveq2 5555	. . 3 |- ( z = B -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
12		df-omul 6476	. . 3 |- .o = ( y e. On , z e. On |-> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) )
13	10, 11, 12	ovmpog 6054	. 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 2164   _Vcvv 2760   (/)c0 3447   |-> cmpt 4091   Oncon0 4395   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   reccrdg 6424   +o coa 6468   .o comu 6469

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-oadd 6475  df-omul 6476

This theorem is referenced by:  om0  6513  omcl  6516  omv2  6520