Definition df-imp 7652

Description: Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7651 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

Assertion

Ref	Expression
df-imp	|- .P. = ( x e. P. , y e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >. )

Distinct variable group:   x, y, q, r, s

Detailed syntax breakdown of Definition df-imp

Step	Hyp	Ref	Expression
1		cmp 7477	. 2 class .P.
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cnp 7474	. . 3 class P.
5		vr	. . . . . . . . . 10 setvar r
6	5	cv 1394	. . . . . . . . 9 class r
7	2	cv 1394	. . . . . . . . . 10 class x
8		c1st 6282	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5317	. . . . . . . . 9 class ( 1st ` x )
10	6, 9	wcel 2200	. . . . . . . 8 wff r e. ( 1st ` x )
11		vs	. . . . . . . . . 10 setvar s
12	11	cv 1394	. . . . . . . . 9 class s
13	3	cv 1394	. . . . . . . . . 10 class y
14	13, 8	cfv 5317	. . . . . . . . 9 class ( 1st ` y )
15	12, 14	wcel 2200	. . . . . . . 8 wff s e. ( 1st ` y )
16		vq	. . . . . . . . . 10 setvar q
17	16	cv 1394	. . . . . . . . 9 class q
18		cmq 7466	. . . . . . . . . 10 class .Q
19	6, 12, 18	co 6000	. . . . . . . . 9 class ( r .Q s )
20	17, 19	wceq 1395	. . . . . . . 8 wff q = ( r .Q s )
21	10, 15, 20	w3a 1002	. . . . . . 7 wff ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) )
22		cnq 7463	. . . . . . 7 class Q.
23	21, 11, 22	wrex 2509	. . . . . 6 wff E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) )
24	23, 5, 22	wrex 2509	. . . . 5 wff E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) )
25	24, 16, 22	crab 2512	. . . 4 class { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) ) }
26		c2nd 6283	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5317	. . . . . . . . 9 class ( 2nd ` x )
28	6, 27	wcel 2200	. . . . . . . 8 wff r e. ( 2nd ` x )
29	13, 26	cfv 5317	. . . . . . . . 9 class ( 2nd ` y )
30	12, 29	wcel 2200	. . . . . . . 8 wff s e. ( 2nd ` y )
31	28, 30, 20	w3a 1002	. . . . . . 7 wff ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) )
32	31, 11, 22	wrex 2509	. . . . . 6 wff E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) )
33	32, 5, 22	wrex 2509	. . . . 5 wff E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) )
34	33, 16, 22	crab 2512	. . . 4 class { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) }
35	25, 34	cop 3669	. . 3 class <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >.
36	2, 3, 4, 4, 35	cmpo 6002	. 2 class ( x e. P. , y e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >. )
37	1, 36	wceq 1395	1 wff .P. = ( x e. P. , y e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) ) } >. )

This definition is referenced by:  mpvlu  7722  dmmp  7724  mulnqprl  7751  mulnqpru  7752  mulclpr  7755  mulnqprlemrl  7756  mulnqprlemru  7757  mulassprg  7764  distrlem1prl  7765  distrlem1pru  7766  distrlem4prl  7767  distrlem4pru  7768  distrlem5prl  7769  distrlem5pru  7770  1idprl  7773  1idpru  7774  recexprlem1ssl  7816  recexprlem1ssu  7817  recexprlemss1l  7818  recexprlemss1u  7819