Definition df-imp 7667

Description: Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7666 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 7492	. 2 class .P.
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cnp 7489	. . 3 class P.
5		vr	. . . . . . . . . 10 setvar r
6	5	cv 1394	. . . . . . . . 9 class r
7	2	cv 1394	. . . . . . . . . 10 class x
8		c1st 6290	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5318	. . . . . . . . 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 5318	. . . . . . . . 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 7481	. . . . . . . . . 10 class .Q
19	6, 12, 18	co 6007	. . . . . . . . 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 7478	. . . . . . 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 6291	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5318	. . . . . . . . 9 class ( 2nd ` x )
28	6, 27	wcel 2200	. . . . . . . 8 wff r e. ( 2nd ` x )
29	13, 26	cfv 5318	. . . . . . . . 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 6009	. 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  7737  dmmp  7739  mulnqprl  7766  mulnqpru  7767  mulclpr  7770  mulnqprlemrl  7771  mulnqprlemru  7772  mulassprg  7779  distrlem1prl  7780  distrlem1pru  7781  distrlem4prl  7782  distrlem4pru  7783  distrlem5prl  7784  distrlem5pru  7785  1idprl  7788  1idpru  7789  recexprlem1ssl  7831  recexprlem1ssu  7832  recexprlemss1l  7833  recexprlemss1u  7834