Theorem mulnnnq0 7660

Description: Multiplication of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)

Assertion

Ref	Expression
mulnnnq0	|- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 ·Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 )

Proof of Theorem mulnnnq0

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

Step	Hyp	Ref	Expression
1		opelxpi 4755	. . . 4 |- ( ( A e. om /\ B e. N. ) -> <. A , B >. e. ( om X. N. ) )
2		enq0ex 7649	. . . . 5 |- ~Q0 e. _V
3	2	ecelqsi 6753	. . . 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 4755	. . . 4 |- ( ( C e. om /\ D e. N. ) -> <. C , D >. e. ( om X. N. ) )
6	2	ecelqsi 6753	. . . 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 2229	. . . 4 |- [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0
10		eqid 2229	. . . 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 2229	. . 3 |- [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0
13		opeq12 3862	. . . . . 6 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14		eceq1 6732	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> [ <. w , v >. ] ~Q0 = [ <. A , B >. ] ~Q0 )
15	14	eqeq2d 2241	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 <-> [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 ) )
16	15	anbi1d 465	. . . . . . 7 |- ( <. w , v >. = <. A , 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		vex 2803	. . . . . . . . . . 11 |- w e. _V
18		vex 2803	. . . . . . . . . . 11 |- v e. _V
19	17, 18	opth 4327	. . . . . . . . . 10 |- ( <. w , v >. = <. A , B >. <-> ( w = A /\ v = B ) )
20		oveq1 6020	. . . . . . . . . . . 12 |- ( w = A -> ( w .o C ) = ( A .o C ) )
21	20	adantr 276	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> ( w .o C ) = ( A .o C ) )
22		oveq1 6020	. . . . . . . . . . . 12 |- ( v = B -> ( v .o D ) = ( B .o D ) )
23	22	adantl 277	. . . . . . . . . . 11 |- ( ( w = A /\ v = B ) -> ( v .o D ) = ( B .o D ) )
24	21, 23	opeq12d 3868	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> <. ( w .o C ) , ( v .o D ) >. = <. ( A .o C ) , ( B .o D ) >. )
25	19, 24	sylbi 121	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> <. ( w .o C ) , ( v .o D ) >. = <. ( A .o C ) , ( B .o D ) >. )
26	25	eceq1d 6733	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 )
27	26	eqeq2d 2241	. . . . . . 7 |- ( <. w , v >. = <. A , B >. -> ( [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 <-> [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) )
28	16, 27	anbi12d 473	. . . . . 6 |- ( <. w , v >. = <. A , B >. -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) ) )
29	13, 28	syl 14	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) ) )
30	29	spc2egv 2894	. . . 4 |- ( ( A e. om /\ B e. N. ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. A , B >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) -> E. w E. v ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) ) )
31		opeq12 3862	. . . . . . 7 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
32		eceq1 6732	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> [ <. u , t >. ] ~Q0 = [ <. C , D >. ] ~Q0 )
33	32	eqeq2d 2241	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> ( [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 <-> [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) )
34	33	anbi2d 464	. . . . . . . 8 |- ( <. u , t >. = <. C , 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 ) ) )
35		vex 2803	. . . . . . . . . . . 12 |- u e. _V
36		vex 2803	. . . . . . . . . . . 12 |- t e. _V
37	35, 36	opth 4327	. . . . . . . . . . 11 |- ( <. u , t >. = <. C , D >. <-> ( u = C /\ t = D ) )
38		oveq2 6021	. . . . . . . . . . . . 13 |- ( u = C -> ( w .o u ) = ( w .o C ) )
39	38	adantr 276	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> ( w .o u ) = ( w .o C ) )
40		oveq2 6021	. . . . . . . . . . . . 13 |- ( t = D -> ( v .o t ) = ( v .o D ) )
41	40	adantl 277	. . . . . . . . . . . 12 |- ( ( u = C /\ t = D ) -> ( v .o t ) = ( v .o D ) )
42	39, 41	opeq12d 3868	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> <. ( w .o u ) , ( v .o t ) >. = <. ( w .o C ) , ( v .o D ) >. )
43	37, 42	sylbi 121	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> <. ( w .o u ) , ( v .o t ) >. = <. ( w .o C ) , ( v .o D ) >. )
44	43	eceq1d 6733	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 )
45	44	eqeq2d 2241	. . . . . . . 8 |- ( <. u , t >. = <. C , D >. -> ( [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 <-> [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) )
46	34, 45	anbi12d 473	. . . . . . 7 |- ( <. u , t >. = <. C , D >. -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) ) )
47	31, 46	syl 14	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) ) )
48	47	spc2egv 2894	. . . . 5 |- ( ( C e. om /\ D e. N. ) -> ( ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. C , D >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o C ) , ( v .o D ) >. ] ~Q0 ) -> E. u E. t ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) )
49	48	2eximdv 1928	. . . 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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) )
50	30, 49	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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( A .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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) )
51	11, 12, 50	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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) )
52		ecexg 6701	. . . 4 |- ( ~Q0 e. _V -> [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 e. _V )
53	2, 52	ax-mp 5	. . 3 |- [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 e. _V
54		eqeq1 2236	. . . . . . . 8 |- ( x = [ <. A , B >. ] ~Q0 -> ( x = [ <. w , v >. ] ~Q0 <-> [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 ) )
55		eqeq1 2236	. . . . . . . 8 |- ( y = [ <. C , D >. ] ~Q0 -> ( y = [ <. u , t >. ] ~Q0 <-> [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) )
56	54, 55	bi2anan9 608	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 ) -> ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) <-> ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) ) )
57		eqeq1 2236	. . . . . . 7 |- ( z = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 -> ( z = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 <-> [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) )
58	56, 57	bi2anan9 608	. . . . . 6 |- ( ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 ) /\ z = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) -> ( ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) )
59	58	3impa 1218	. . . . 5 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) -> ( ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , t >. ] ~Q0 ) /\ z = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) <-> ( ( [ <. A , B >. ] ~Q0 = [ <. w , v >. ] ~Q0 /\ [ <. C , D >. ] ~Q0 = [ <. u , t >. ] ~Q0 ) /\ [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) )
60	59	4exbidv 1916	. . . 4 |- ( ( x = [ <. A , B >. ] ~Q0 /\ y = [ <. C , D >. ] ~Q0 /\ z = [ <. ( A .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 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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) ) )
61		mulnq0mo 7658	. . . 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 u ) , ( v .o t ) >. ] ~Q0 ) )
62		dfmq0qs 7639	. . . 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 u ) , ( v .o t ) >. ] ~Q0 ) ) }
63	60, 61, 62	ovig 6138	. . 3 |- ( ( [ <. A , B >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) /\ [ <. C , D >. ] ~Q0 e. ( ( om X. N. ) /. ~Q0 ) /\ [ <. ( A .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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) -> ( [ <. A , B >. ] ~Q0 ·Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) )
64	53, 63	mp3an3 1360	. 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 C ) , ( B .o D ) >. ] ~Q0 = [ <. ( w .o u ) , ( v .o t ) >. ] ~Q0 ) -> ( [ <. A , B >. ] ~Q0 ·Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 ) )
65	8, 51, 64	sylc 62	1 |- ( ( ( A e. om /\ B e. N. ) /\ ( C e. om /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q0 ·Q0 [ <. C , D >. ] ~Q0 ) = [ <. ( A .o C ) , ( B .o D ) >. ] ~Q0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2800   <.cop 3670   omcom 4686   X. cxp 4721  (class class class)co 6013   .o comu 6575   [cec 6695   /.cqs 6696   N.cnpi 7482   ~_Q0 ceq0 7496   ·_Q0 cmq0 7500

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-enq0 7634  df-nq0 7635  df-mq0 7638

This theorem is referenced by:  mulclnq0  7662  nqnq0m  7665  nq0m0r  7666  distrnq0  7669  mulcomnq0  7670  nq02m  7675