Theorem onmsuc 7609

Description: Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Assertion

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

Proof of Theorem onmsuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7086	. . . . 5 |- ( B e. om -> suc B e. om )
2		nnon 7071	. . . . 5 |- ( suc B e. om -> suc B e. On )
3	1, 2	syl 17	. . . 4 |- ( B e. om -> suc B e. On )
4		omv 7592	. . . 4 |- ( ( 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	. . 3 |- ( ( A e. On /\ B e. om ) -> ( A .o suc B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) )
6	1	adantl 482	. . . 4 |- ( ( A e. On /\ B e. om ) -> suc B e. om )
7		fvres 6207	. . . 4 |- ( suc B e. om -> ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` suc B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) )
8	6, 7	syl 17	. . 3 |- ( ( A e. On /\ B e. om ) -> ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` suc B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` suc B ) )
9	5, 8	eqtr4d 2659	. 2 |- ( ( A e. On /\ B e. om ) -> ( A .o suc B ) = ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` suc B ) )
10		ovex 6678	. . . . 5 |- ( A .o B ) e. _V
11		oveq1 6657	. . . . . 6 |- ( x = ( A .o B ) -> ( x +o A ) = ( ( A .o B ) +o A ) )
12		eqid 2622	. . . . . 6 |- ( x e. _V |-> ( x +o A ) ) = ( x e. _V |-> ( x +o A ) )
13		ovex 6678	. . . . . 6 |- ( ( A .o B ) +o A ) e. _V
14	11, 12, 13	fvmpt 6282	. . . . 5 |- ( ( A .o B ) e. _V -> ( ( x e. _V |-> ( x +o A ) ) ` ( A .o B ) ) = ( ( A .o B ) +o A ) )
15	10, 14	ax-mp 5	. . . 4 |- ( ( x e. _V |-> ( x +o A ) ) ` ( A .o B ) ) = ( ( A .o B ) +o A )
16		nnon 7071	. . . . . . 7 |- ( B e. om -> B e. On )
17		omv 7592	. . . . . . 7 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
18	16, 17	sylan2 491	. . . . . 6 |- ( ( A e. On /\ B e. om ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
19		fvres 6207	. . . . . . 7 |- ( B e. om -> ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
20	19	adantl 482	. . . . . 6 |- ( ( A e. On /\ B e. om ) -> ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
21	18, 20	eqtr4d 2659	. . . . 5 |- ( ( A e. On /\ B e. om ) -> ( A .o B ) = ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) )
22	21	fveq2d 6195	. . . 4 |- ( ( A e. On /\ B e. om ) -> ( ( x e. _V |-> ( x +o A ) ) ` ( A .o B ) ) = ( ( x e. _V |-> ( x +o A ) ) ` ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) ) )
23	15, 22	syl5eqr 2670	. . 3 |- ( ( A e. On /\ B e. om ) -> ( ( A .o B ) +o A ) = ( ( x e. _V |-> ( x +o A ) ) ` ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) ) )
24		frsuc 7532	. . . 4 |- ( B e. om -> ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` suc B ) = ( ( x e. _V |-> ( x +o A ) ) ` ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) ) )
25	24	adantl 482	. . 3 |- ( ( A e. On /\ B e. om ) -> ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` suc B ) = ( ( x e. _V |-> ( x +o A ) ) ` ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` B ) ) )
26	23, 25	eqtr4d 2659	. 2 |- ( ( A e. On /\ B e. om ) -> ( ( A .o B ) +o A ) = ( ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) |` om ) ` suc B ) )
27	9, 26	eqtr4d 2659	1 |- ( ( A e. On /\ B e. om ) -> ( 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   |` cres 5116   Oncon0 5723   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-omul 7565

This theorem is referenced by:  om1  7622  nnmsuc  7687