Theorem omeu 7665

Description: The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
omeu	|- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> E! z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )

Distinct variable groups:   x, A, y, z   x, B, y, z

Proof of Theorem omeu

Dummy variables r s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omeulem1 7662	. . 3 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> E. x e. On E. y e. A ( ( A .o x ) +o y ) = B )
2		opex 4932	. . . . . . . . 9 |- <. x , y >. e. _V
3	2	isseti 3209	. . . . . . . 8 |- E. z z = <. x , y >.
4		19.41v 1914	. . . . . . . 8 |- ( E. z ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> ( E. z z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
5	3, 4	mpbiran 953	. . . . . . 7 |- ( E. z ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> ( ( A .o x ) +o y ) = B )
6	5	rexbii 3041	. . . . . 6 |- ( E. y e. A E. z ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> E. y e. A ( ( A .o x ) +o y ) = B )
7		rexcom4 3225	. . . . . 6 |- ( E. y e. A E. z ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> E. z E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
8	6, 7	bitr3i 266	. . . . 5 |- ( E. y e. A ( ( A .o x ) +o y ) = B <-> E. z E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
9	8	rexbii 3041	. . . 4 |- ( E. x e. On E. y e. A ( ( A .o x ) +o y ) = B <-> E. x e. On E. z E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
10		rexcom4 3225	. . . 4 |- ( E. x e. On E. z E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> E. z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
11	9, 10	bitri 264	. . 3 |- ( E. x e. On E. y e. A ( ( A .o x ) +o y ) = B <-> E. z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
12	1, 11	sylib 208	. 2 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> E. z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )
13		simp2rl 1130	. . . . . . . . . . 11 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> z = <. x , y >. )
14		simp3rl 1134	. . . . . . . . . . . 12 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> t = <. r , s >. )
15		simp2rr 1131	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> ( ( A .o x ) +o y ) = B )
16		simp3rr 1135	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> ( ( A .o r ) +o s ) = B )
17	15, 16	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> ( ( A .o x ) +o y ) = ( ( A .o r ) +o s ) )
18		simp11 1091	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> A e. On )
19		simp13 1093	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> A =/= (/) )
20		simp2ll 1128	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> x e. On )
21		simp2lr 1129	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> y e. A )
22		simp3ll 1132	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> r e. On )
23		simp3lr 1133	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> s e. A )
24		omopth2 7664	. . . . . . . . . . . . . . 15 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( x e. On /\ y e. A ) /\ ( r e. On /\ s e. A ) ) -> ( ( ( A .o x ) +o y ) = ( ( A .o r ) +o s ) <-> ( x = r /\ y = s ) ) )
25	18, 19, 20, 21, 22, 23, 24	syl222anc 1342	. . . . . . . . . . . . . 14 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> ( ( ( A .o x ) +o y ) = ( ( A .o r ) +o s ) <-> ( x = r /\ y = s ) ) )
26	17, 25	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> ( x = r /\ y = s ) )
27		opeq12 4404	. . . . . . . . . . . . 13 |- ( ( x = r /\ y = s ) -> <. x , y >. = <. r , s >. )
28	26, 27	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> <. x , y >. = <. r , s >. )
29	14, 28	eqtr4d 2659	. . . . . . . . . . 11 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> t = <. x , y >. )
30	13, 29	eqtr4d 2659	. . . . . . . . . 10 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) /\ ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) ) -> z = t )
31	30	3expia 1267	. . . . . . . . 9 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) ) -> ( ( ( r e. On /\ s e. A ) /\ ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) -> z = t ) )
32	31	exp4b 632	. . . . . . . 8 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> ( ( ( x e. On /\ y e. A ) /\ ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) -> ( ( r e. On /\ s e. A ) -> ( ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) -> z = t ) ) ) )
33	32	expd 452	. . . . . . 7 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> ( ( x e. On /\ y e. A ) -> ( ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) -> ( ( r e. On /\ s e. A ) -> ( ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) -> z = t ) ) ) ) )
34	33	rexlimdvv 3037	. . . . . 6 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> ( E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) -> ( ( r e. On /\ s e. A ) -> ( ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) -> z = t ) ) ) )
35	34	imp 445	. . . . 5 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) -> ( ( r e. On /\ s e. A ) -> ( ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) -> z = t ) ) )
36	35	rexlimdvv 3037	. . . 4 |- ( ( ( A e. On /\ B e. On /\ A =/= (/) ) /\ E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) -> ( E. r e. On E. s e. A ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) -> z = t ) )
37	36	expimpd 629	. . 3 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> ( ( E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) /\ E. r e. On E. s e. A ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) -> z = t ) )
38	37	alrimivv 1856	. 2 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> A. z A. t ( ( E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) /\ E. r e. On E. s e. A ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) -> z = t ) )
39		opeq1 4402	. . . . . . 7 |- ( x = r -> <. x , y >. = <. r , y >. )
40	39	eqeq2d 2632	. . . . . 6 |- ( x = r -> ( z = <. x , y >. <-> z = <. r , y >. ) )
41		oveq2 6658	. . . . . . . 8 |- ( x = r -> ( A .o x ) = ( A .o r ) )
42	41	oveq1d 6665	. . . . . . 7 |- ( x = r -> ( ( A .o x ) +o y ) = ( ( A .o r ) +o y ) )
43	42	eqeq1d 2624	. . . . . 6 |- ( x = r -> ( ( ( A .o x ) +o y ) = B <-> ( ( A .o r ) +o y ) = B ) )
44	40, 43	anbi12d 747	. . . . 5 |- ( x = r -> ( ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> ( z = <. r , y >. /\ ( ( A .o r ) +o y ) = B ) ) )
45		opeq2 4403	. . . . . . 7 |- ( y = s -> <. r , y >. = <. r , s >. )
46	45	eqeq2d 2632	. . . . . 6 |- ( y = s -> ( z = <. r , y >. <-> z = <. r , s >. ) )
47		oveq2 6658	. . . . . . 7 |- ( y = s -> ( ( A .o r ) +o y ) = ( ( A .o r ) +o s ) )
48	47	eqeq1d 2624	. . . . . 6 |- ( y = s -> ( ( ( A .o r ) +o y ) = B <-> ( ( A .o r ) +o s ) = B ) )
49	46, 48	anbi12d 747	. . . . 5 |- ( y = s -> ( ( z = <. r , y >. /\ ( ( A .o r ) +o y ) = B ) <-> ( z = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) )
50	44, 49	cbvrex2v 3180	. . . 4 |- ( E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> E. r e. On E. s e. A ( z = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) )
51		eqeq1 2626	. . . . . 6 |- ( z = t -> ( z = <. r , s >. <-> t = <. r , s >. ) )
52	51	anbi1d 741	. . . . 5 |- ( z = t -> ( ( z = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) <-> ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) )
53	52	2rexbidv 3057	. . . 4 |- ( z = t -> ( E. r e. On E. s e. A ( z = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) <-> E. r e. On E. s e. A ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) )
54	50, 53	syl5bb 272	. . 3 |- ( z = t -> ( E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> E. r e. On E. s e. A ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) )
55	54	eu4 2518	. 2 |- ( E! z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) <-> ( E. z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) /\ A. z A. t ( ( E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) /\ E. r e. On E. s e. A ( t = <. r , s >. /\ ( ( A .o r ) +o s ) = B ) ) -> z = t ) ) )
56	12, 38, 55	sylanbrc 698	1 |- ( ( A e. On /\ B e. On /\ A =/= (/) ) -> E! z E. x e. On E. y e. A ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   =/= wne 2794   E.wrex 2913   (/)c0 3915   <.cop 4183   Oncon0 5723  (class class class)co 6650   +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-rep 4771  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-rmo 2920  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-int 4476  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565

This theorem is referenced by:  oeeui  7682  omxpenlem  8061