Theorem omsuc 7606

Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
omsuc	|- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )

Proof of Theorem omsuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgsuc 7520	. . 3 |- ( B e. On -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) = ( ( x e. _V |-> ( x +o A ) ) ` ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) ) )
2	1	adantl 482	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) = ( ( x e. _V |-> ( x +o A ) ) ` ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) ) )
3		suceloni 7013	. . 3 |- ( B e. On -> suc B e. On )
4		omv 7592	. . 3 |- ( ( A e. On /\ suc B e. On ) -> ( A .o suc B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) )
5	3, 4	sylan2 491	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) )
6		ovex 6678	. . . 4 |- ( A .o B ) e. _V
7		oveq1 6657	. . . . 5 |- ( x = ( A .o B ) -> ( x +o A ) = ( ( A .o B ) +o A ) )
8		eqid 2622	. . . . 5 |- ( x e. _V |-> ( x +o A ) ) = ( x e. _V |-> ( x +o A ) )
9		ovex 6678	. . . . 5 |- ( ( A .o B ) +o A ) e. _V
10	7, 8, 9	fvmpt 6282	. . . 4 |- ( ( A .o B ) e. _V -> ( ( x e. _V |-> ( x +o A ) ) ` ( A .o B ) ) = ( ( A .o B ) +o A ) )
11	6, 10	ax-mp 5	. . 3 |- ( ( x e. _V |-> ( x +o A ) ) ` ( A .o B ) ) = ( ( A .o B ) +o A )
12		omv 7592	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
13	12	fveq2d 6195	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( x e. _V |-> ( x +o A ) ) ` ( A .o B ) ) = ( ( x e. _V |-> ( x +o A ) ) ` ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) ) )
14	11, 13	syl5eqr 2670	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( A .o B ) +o A ) = ( ( x e. _V |-> ( x +o A ) ) ` ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) ) )
15	2, 5, 14	3eqtr4d 2666	1 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   Oncon0 5723   suc csuc 5725   ` cfv 5888  (class class class)co 6650   reccrdg 7505   +o coa 7557   .o comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-omul 7565

This theorem is referenced by:  omcl  7616  om0r  7619  om1r  7623  omordi  7646  omwordri  7652  omlimcl  7658  odi  7659  omass  7660  oneo  7661  omeulem1  7662  omeulem2  7663  oeoelem  7678  oaabs2  7725  omxpenlem  8061  cantnflt  8569  cantnflem1d  8585  infxpenc  8841