Theorem omsuc 6074

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		df-suc 4126	. . . . . . 7 |- suc B = ( B u. { B } )
2		iuneq1 3691	. . . . . . 7 |- ( suc B = ( B u. { B } ) -> U_ x e. suc B ( ( A .o x ) +o A ) = U_ x e. ( B u. { B } ) ( ( A .o x ) +o A ) )
3	1, 2	ax-mp 7	. . . . . 6 |- U_ x e. suc B ( ( A .o x ) +o A ) = U_ x e. ( B u. { B } ) ( ( A .o x ) +o A )
4		iunxun 3756	. . . . . 6 |- U_ x e. ( B u. { B } ) ( ( A .o x ) +o A ) = ( U_ x e. B ( ( A .o x ) +o A ) u. U_ x e. { B } ( ( A .o x ) +o A ) )
5	3, 4	eqtri 2101	. . . . 5 |- U_ x e. suc B ( ( A .o x ) +o A ) = ( U_ x e. B ( ( A .o x ) +o A ) u. U_ x e. { B } ( ( A .o x ) +o A ) )
6		oveq2 5540	. . . . . . . 8 |- ( x = B -> ( A .o x ) = ( A .o B ) )
7	6	oveq1d 5547	. . . . . . 7 |- ( x = B -> ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
8	7	iunxsng 3753	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
9	8	uneq2d 3126	. . . . 5 |- ( B e. On -> ( U_ x e. B ( ( A .o x ) +o A ) u. U_ x e. { B } ( ( A .o x ) +o A ) ) = ( U_ x e. B ( ( A .o x ) +o A ) u. ( ( A .o B ) +o A ) ) )
10	5, 9	syl5eq 2125	. . . 4 |- ( B e. On -> U_ x e. suc B ( ( A .o x ) +o A ) = ( U_ x e. B ( ( A .o x ) +o A ) u. ( ( A .o B ) +o A ) ) )
11	10	adantl 271	. . 3 |- ( ( A e. On /\ B e. On ) -> U_ x e. suc B ( ( A .o x ) +o A ) = ( U_ x e. B ( ( A .o x ) +o A ) u. ( ( A .o B ) +o A ) ) )
12		suceloni 4245	. . . 4 |- ( B e. On -> suc B e. On )
13		omv2 6068	. . . 4 |- ( ( A e. On /\ suc B e. On ) -> ( A .o suc B ) = U_ x e. suc B ( ( A .o x ) +o A ) )
14	12, 13	sylan2 280	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = U_ x e. suc B ( ( A .o x ) +o A ) )
15		omv2 6068	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) )
16	15	uneq1d 3125	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( A .o B ) u. ( ( A .o B ) +o A ) ) = ( U_ x e. B ( ( A .o x ) +o A ) u. ( ( A .o B ) +o A ) ) )
17	11, 14, 16	3eqtr4d 2123	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) u. ( ( A .o B ) +o A ) ) )
18		omcl 6064	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )
19		simpl 107	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
20		oaword1 6073	. . . 4 |- ( ( ( A .o B ) e. On /\ A e. On ) -> ( A .o B ) C_ ( ( A .o B ) +o A ) )
21		ssequn1 3142	. . . 4 |- ( ( A .o B ) C_ ( ( A .o B ) +o A ) <-> ( ( A .o B ) u. ( ( A .o B ) +o A ) ) = ( ( A .o B ) +o A ) )
22	20, 21	sylib 120	. . 3 |- ( ( ( A .o B ) e. On /\ A e. On ) -> ( ( A .o B ) u. ( ( A .o B ) +o A ) ) = ( ( A .o B ) +o A ) )
23	18, 19, 22	syl2anc 403	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( A .o B ) u. ( ( A .o B ) +o A ) ) = ( ( A .o B ) +o A ) )
24	17, 23	eqtrd 2113	1 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   u. cun 2971   C_ wss 2973   {csn 3398   U_ciun 3678   Oncon0 4118   suc csuc 4120  (class class class)co 5532   +o coa 6021   .o comu 6022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029

This theorem is referenced by:  onmsuc  6075