Theorem omsuc 6745

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 4516	. . . . . . 7 |- suc B = ( B u. { B } )
2		iuneq1 4025	. . . . . . 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 4092	. . . . . 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 2259	. . . . 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 6093	. . . . . . . 8 |- ( x = B -> ( A .o x ) = ( A .o B ) )
7	6	oveq1d 6100	. . . . . . 7 |- ( x = B -> ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
8	7	iunxsng 4088	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
9	8	uneq2d 3383	. . . . 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	eqtrid 2283	. . . 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 277	. . 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		onsuc 4648	. . . 4 |- ( B e. On -> suc B e. On )
13		omv2 6738	. . . 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 286	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = U_ x e. suc B ( ( A .o x ) +o A ) )
15		omv2 6738	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) )
16	15	uneq1d 3382	. . 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 2281	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) u. ( ( A .o B ) +o A ) ) )
18		omcl 6734	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )
19		simpl 109	. . 3 |- ( ( A e. On /\ B e. On ) -> A e. On )
20		oaword1 6744	. . . 4 |- ( ( ( A .o B ) e. On /\ A e. On ) -> ( A .o B ) C_ ( ( A .o B ) +o A ) )
21		ssequn1 3399	. . . 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 122	. . 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 415	. 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 2271	1 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   u. cun 3218   C_ wss 3220   {csn 3709   U_ciun 4012   Oncon0 4508   suc csuc 4510  (class class class)co 6085   +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:  onmsuc  6746