Theorem omsuc 6440

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 4349	. . . . . . 7 |- suc B = ( B u. { B } )
2		iuneq1 3879	. . . . . . 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 3945	. . . . . 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 2186	. . . . 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 5850	. . . . . . . 8 |- ( x = B -> ( A .o x ) = ( A .o B ) )
7	6	oveq1d 5857	. . . . . . 7 |- ( x = B -> ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
8	7	iunxsng 3941	. . . . . 6 |- ( B e. On -> U_ x e. { B } ( ( A .o x ) +o A ) = ( ( A .o B ) +o A ) )
9	8	uneq2d 3276	. . . . 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 2211	. . . 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 4478	. . . 4 |- ( B e. On -> suc B e. On )
13		omv2 6433	. . . 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 6433	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) = U_ x e. B ( ( A .o x ) +o A ) )
16	15	uneq1d 3275	. . 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 2208	. 2 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) u. ( ( A .o B ) +o A ) ) )
18		omcl 6429	. . 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 6439	. . . 4 |- ( ( ( A .o B ) e. On /\ A e. On ) -> ( A .o B ) C_ ( ( A .o B ) +o A ) )
21		ssequn1 3292	. . . 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 2198	1 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   u. cun 3114   C_ wss 3116   {csn 3576   U_ciun 3866   Oncon0 4341   suc csuc 4343  (class class class)co 5842   +o coa 6381   .o comu 6382

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389

This theorem is referenced by:  onmsuc  6441