Theorem omsuc 6416

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 4331	. . . . . . 7 |- suc B = ( B u. { B } )
2		iuneq1 3862	. . . . . . 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 5	. . . . . 6 |- U_ x e. suc B ( ( A .o x ) +o A ) = U_ x e. ( B u. { B } ) ( ( A .o x ) +o A )
4		iunxun 3928	. . . . . 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 2178	. . . . 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 5829	. . . . . . . 8 |- ( x = B -> ( A .o x ) = ( A .o B ) )
7	6	oveq1d 5836	. . . . . . 7 |- ( x = B -> ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
8	7	iunxsng 3924	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
9	8	uneq2d 3261	. . . . 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 2202	. . . 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 275	. . 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 4459	. . . 4 |- ( B e. On -> suc B e. On )
13		omv2 6409	. . . 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 284	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = U_ x e. suc B ( ( A .o x ) +o A ) )
15		omv2 6409	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) )
16	15	uneq1d 3260	. . 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 2200	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) u. ( ( A .o B ) +o A ) ) )
18		omcl 6405	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )
19		simpl 108	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
20		oaword1 6415	. . . 4 |- ( ( ( A .o B ) e. On /\ A e. On ) -> ( A .o B ) C_ ( ( A .o B ) +o A ) )
21		ssequn1 3277	. . . 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 121	. . 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 409	. 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 2190	1 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   u. cun 3100   C_ wss 3102   {csn 3560   U_ciun 3849   Oncon0 4323   suc csuc 4325  (class class class)co 5821   +o coa 6357   .o comu 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-oadd 6364  df-omul 6365

This theorem is referenced by:  onmsuc  6417