Theorem addnnnq0 7439

Description: Addition of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)

Assertion

Ref	Expression
addnnnq0	|- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 +Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 )

Proof of Theorem addnnnq0

Dummy variables x y z w v u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 4655	. . . 4 |- ( ( A e. om /\ B e. N. ) -> <. A , B >. e. ( om X. N. ) )
2		enq0ex 7429	. . . . 5 |- ~Q0 e. _V
3	2	ecelqsi 6583	. . . 4 |- ( <. A , B >. e. ( om X. N. ) -> [ <. A , B >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
4	1, 3	syl 14	. . 3 |- ( ( A e. om /\ B e. N. ) -> [ <. A , B >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
5		opelxpi 4655	. . . 4 |- ( ( C e. om /\ D e. N. ) -> <. C , D >. e. ( om X. N. ) )
6	2	ecelqsi 6583	. . . 4 |- ( <. C , D >. e. ( om X. N. ) -> [ <. C , D >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
7	5, 6	syl 14	. . 3 |- ( ( C e. om /\ D e. N. ) -> [ <. C , D >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) )
8	4, 7	anim12i 338	. 2 |- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) /\ [ <. C , D >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) ) )
9		eqid 2177	. . . 4 |- [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0
10		eqid 2177	. . . 4 |- [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0
11	9, 10	pm3.2i 272	. . 3 |- ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 )
12		eqid 2177	. . 3 |- [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0
13		opeq12 3778	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6565	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~Q0 = [ <. A , B >. ] ~Q0 )
15	14	eqeq2d 2189	. . . . . . 7 |- ( ( w = A /\ v = B ) -> ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 <-> [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 ) )
16	15	anbi1d 465	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) <-> ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) ) )
17		simpl 109	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> w = A )
18	17	oveq1d 5884	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .o D ) = ( A .o D ) )
19		simpr 110	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 5884	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .o C ) = ( B .o C ) )
21	18, 20	oveq12d 5887	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .o D ) +o ( v .o C ) ) = ( ( A .o D ) +o ( B .o C ) ) )
22	19	oveq1d 5884	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( v .o D ) = ( B .o D ) )
23	21, 22	opeq12d 3784	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. = <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. )
24	23	eceq1d 6565	. . . . . . 7 |- ( ( w = A /\ v = B ) -> [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 )
25	24	eqeq2d 2189	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 <-> [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) )
26	16, 25	anbi12d 473	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) ) )
27	26	spc2egv 2827	. . . 4 |- ( ( A e. om /\ B e. N. ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> E. w E. v ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 ) ) )
28		opeq12 3778	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
29	28	eceq1d 6565	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~Q0 = [ <. C , D >. ] ~Q0 )
30	29	eqeq2d 2189	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> ( [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 <-> [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) )
31	30	anbi2d 464	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) <-> ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) ) )
32		simpr 110	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> t = D )
33	32	oveq2d 5885	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .o t ) = ( w .o D ) )
34		simpl 109	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> u = C )
35	34	oveq2d 5885	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .o u ) = ( v .o C ) )
36	33, 35	oveq12d 5887	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .o t ) +o ( v .o u ) ) = ( ( w .o D ) +o ( v .o C ) ) )
37	32	oveq2d 5885	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( v .o t ) = ( v .o D ) )
38	36, 37	opeq12d 3784	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. = <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. )
39	38	eceq1d 6565	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 )
40	39	eqeq2d 2189	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 <-> [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 ) )
41	31, 40	anbi12d 473	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 ) ) )
42	41	spc2egv 2827	. . . . 5 |- ( ( C e. om /\ D e. N. ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 ) -> E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) )
43	42	2eximdv 1882	. . . 4 |- ( ( C e. om /\ D e. N. ) -> ( E. w E. v ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o D ) +o ( v .o C ) ) , ( v .o D ) >. ] ~Q0 ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) )
44	27, 43	sylan9 409	. . 3 |- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) )
45	11, 12, 44	mp2ani 432	. 2 |- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) )
46		ecexg 6533	. . . 4 |- ( ~Q0 e. _V -> [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 e. _V )
47	2, 46	ax-mp 5	. . 3 |- [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 e. _V
48		simp1 997	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> x = [ <. A , B >. ] ~Q0 )
49	48	eqeq1d 2186	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> ( x = [ <. w , v >. ] ~Q0 <-> [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 ) )
50		simp2 998	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> y = [ <. C , D >. ] ~Q0 )
51	50	eqeq1d 2186	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> ( y = [ <. u , t >. ] ~Q0 <-> [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) )
52	49, 51	anbi12d 473	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) <-> ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) ) )
53		simp3 999	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 )
54	53	eqeq1d 2186	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> ( z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 <-> [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) )
55	52, 54	anbi12d 473	. . . . 5 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> ( ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) )
56	55	4exbidv 1870	. . . 4 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) -> ( E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) <-> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) )
57		addnq0mo 7437	. . . 4 |- ( ( x e. ( ( om X. N. ) /. ~Q0 ) /\ y e. ( ( om X. N. ) /. ~Q0 ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) )
58		dfplq0qs 7420	. . . 4 |- +Q0 = { <. <. x , y >. , z >. | ( ( x e. ( ( om X. N. ) /. ~Q0 ) /\ y e. ( ( om X. N. ) /. ~Q0 ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) ) }
59	56, 57, 58	ovig 5990	. . 3 |- ( ( [ <. A , B >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) /\ [ <. C , D >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 e. _V ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) -> ( [ <. A , B >. ] ~Q0 +Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) )
60	47, 59	mp3an3 1326	. 2 |- ( ( [ <. A , B >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) /\ [ <. C , D >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 = [ <. ( ( w .o t ) +o ( v .o u ) ) , ( v .o t ) >. ] ~Q0 ) -> ( [ <. A , B >. ] ~Q0 +Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 ) )
61	8, 45, 60	sylc 62	1 |- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 +Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( ( A .o D ) +o ( B .o C ) ) , ( B .o D ) >. ] ~Q0 )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   <.cop 3594   omcom 4586   X. cxp 4621  (class class class)co 5869   +o coa 6408   .o comu 6409   [cec 6527   /.cqs 6528   N.cnpi 7262   ~_Q0 ceq0 7276   +_Q0 cplq0 7279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-mi 7296  df-enq0 7414  df-nq0 7415  df-plq0 7417

This theorem is referenced by:  addclnq0  7441  nqpnq0nq  7443  nqnq0a  7444  nq0a0  7447  nnanq0  7448  distrnq0  7449  addassnq0  7452