Theorem omv 6728

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 4537	. . 3 |- (/) e. On
2		omfnex 6722	. . . 4 |- ( A e. On -> ( x e. _V |-> ( x +o A ) ) Fn _V )
3		rdgexggg 6648	. . . 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 1311	. . 3 |- ( ( A e. On /\ (/) e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V )
5	1, 4	mp3an2 1366	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V )
6		oveq2 6093	. . . . . 6 |- ( y = A -> ( x +o y ) = ( x +o A ) )
7	6	mpteq2dv 4222	. . . . 5 |- ( y = A -> ( x e. _V |-> ( x +o y ) ) = ( x e. _V |-> ( x +o A ) ) )
8		rdgeq1 6642	. . . . 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 5697	. . 3 |- ( y = A -> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) )
11		fveq2 5695	. . 3 |- ( z = B -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
12		df-omul 6692	. . 3 |- .o = ( y e. On , z e. On |-> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) )
13	10, 11, 12	ovmpog 6223	. 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 1379	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   |-> cmpt 4192   Oncon0 4508   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   reccrdg 6640   +o coa 6684   .o comu 6685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-oadd 6691  df-omul 6692

This theorem is used by:  om0  6731  omcl  6734  omv2  6738