Theorem addnnnq0 7509

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 4691	. . . 4 |- ( ( A e. om /\ B e. N. ) -> <. A , B >. e. ( om X. N. ) )
2		enq0ex 7499	. . . . 5 |- ~Q0 e. _V
3	2	ecelqsi 6643	. . . 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 4691	. . . 4 |- ( ( C e. om /\ D e. N. ) -> <. C , D >. e. ( om X. N. ) )
6	2	ecelqsi 6643	. . . 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 2193	. . . 4 |- [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0
10		eqid 2193	. . . 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 2193	. . 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 3806	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6623	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~Q0 = [ <. A , B >. ] ~Q0 )
15	14	eqeq2d 2205	. . . . . . 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 5933	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( w .o D ) = ( A .o D ) )
19		simpr 110	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 5933	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> ( v .o C ) = ( B .o C ) )
21	18, 20	oveq12d 5936	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( ( w .o D ) +o ( v .o C ) ) = ( ( A .o D ) +o ( B .o C ) ) )
22	19	oveq1d 5933	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( v .o D ) = ( B .o D ) )
23	21, 22	opeq12d 3812	. . . . . . . 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 6623	. . . . . . 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 2205	. . . . . 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 2850	. . . 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 3806	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
29	28	eceq1d 6623	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~Q0 = [ <. C , D >. ] ~Q0 )
30	29	eqeq2d 2205	. . . . . . . 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 5934	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( w .o t ) = ( w .o D ) )
34		simpl 109	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> u = C )
35	34	oveq2d 5934	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> ( v .o u ) = ( v .o C ) )
36	33, 35	oveq12d 5936	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( ( w .o t ) +o ( v .o u ) ) = ( ( w .o D ) +o ( v .o C ) ) )
37	32	oveq2d 5934	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( v .o t ) = ( v .o D ) )
38	36, 37	opeq12d 3812	. . . . . . . . 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 6623	. . . . . . . 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 2205	. . . . . . 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 2850	. . . . 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 1893	. . . 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 6591	. . . 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 999	. . . . . . . 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 2202	. . . . . . 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 1000	. . . . . . . 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 2202	. . . . . . 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 1001	. . . . . . 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 2202	. . . . . 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 1881	. . . 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 7507	. . . 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 7490	. . . 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 6040	. . 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 1337	. 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 980   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   <.cop 3621   omcom 4622   X. cxp 4657  (class class class)co 5918   +o coa 6466   .o comu 6467   [cec 6585   /.cqs 6586   N.cnpi 7332   ~_Q0 ceq0 7346   +_Q0 cplq0 7349

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-enq0 7484  df-nq0 7485  df-plq0 7487

This theorem is referenced by:  addclnq0  7511  nqpnq0nq  7513  nqnq0a  7514  nq0a0  7517  nnanq0  7518  distrnq0  7519  addassnq0  7522