Definition df-imp 7481

Description: Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7480 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 7306	. 2 class .P.
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cnp 7303	. . 3 class P.
5		vr	. . . . . . . . . 10 setvar r
6	5	cv 1362	. . . . . . . . 9 class r
7	2	cv 1362	. . . . . . . . . 10 class x
8		c1st 6152	. . . . . . . . . 10 class 1st
9	7, 8	cfv 5228	. . . . . . . . 9 class ( 1st ` x )
10	6, 9	wcel 2158	. . . . . . . 8 wff r e. ( 1st ` x )
11		vs	. . . . . . . . . 10 setvar s
12	11	cv 1362	. . . . . . . . 9 class s
13	3	cv 1362	. . . . . . . . . 10 class y
14	13, 8	cfv 5228	. . . . . . . . 9 class ( 1st ` y )
15	12, 14	wcel 2158	. . . . . . . 8 wff s e. ( 1st ` y )
16		vq	. . . . . . . . . 10 setvar q
17	16	cv 1362	. . . . . . . . 9 class q
18		cmq 7295	. . . . . . . . . 10 class .Q
19	6, 12, 18	co 5888	. . . . . . . . 9 class ( r .Q s )
20	17, 19	wceq 1363	. . . . . . . 8 wff q = ( r .Q s )
21	10, 15, 20	w3a 979	. . . . . . 7 wff ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) )
22		cnq 7292	. . . . . . 7 class Q.
23	21, 11, 22	wrex 2466	. . . . . 6 wff E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r .Q s ) )
24	23, 5, 22	wrex 2466	. . . . 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 2469	. . . 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 6153	. . . . . . . . . 10 class 2nd
27	7, 26	cfv 5228	. . . . . . . . 9 class ( 2nd ` x )
28	6, 27	wcel 2158	. . . . . . . 8 wff r e. ( 2nd ` x )
29	13, 26	cfv 5228	. . . . . . . . 9 class ( 2nd ` y )
30	12, 29	wcel 2158	. . . . . . . 8 wff s e. ( 2nd ` y )
31	28, 30, 20	w3a 979	. . . . . . 7 wff ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) )
32	31, 11, 22	wrex 2466	. . . . . 6 wff E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r .Q s ) )
33	32, 5, 22	wrex 2466	. . . . 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 2469	. . . 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 3607	. . 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 5890	. 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 1363	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  7551  dmmp  7553  mulnqprl  7580  mulnqpru  7581  mulclpr  7584  mulnqprlemrl  7585  mulnqprlemru  7586  mulassprg  7593  distrlem1prl  7594  distrlem1pru  7595  distrlem4prl  7596  distrlem4pru  7597  distrlem5prl  7598  distrlem5pru  7599  1idprl  7602  1idpru  7603  recexprlem1ssl  7645  recexprlem1ssu  7646  recexprlemss1l  7647  recexprlemss1u  7648