Theorem dfplpq2 7103

Description: Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)

Assertion

Ref	Expression
dfplpq2	|- +pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) }

Distinct variable group:   x, y, z, w, v, u, f

Proof of Theorem dfplpq2

Step	Hyp	Ref	Expression
1		df-mpo 5731	. 2 |- ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) }
2		df-plpq 7093	. 2 |- +pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
3		1st2nd2 6024	. . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
4	3	eqeq1d 2121	. . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5		1st2nd2 6024	. . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
6	5	eqeq1d 2121	. . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
7	4, 6	bi2anan9 578	. . . . . . . 8 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) ) )
8	7	anbi1d 458	. . . . . . 7 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) <-> ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) ) )
9		xp1st 6014	. . . . . . . . . . . . . 14 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
10	9	ad2antlr 478	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 1st ` y ) e. N. )
11	7	biimpa 292	. . . . . . . . . . . . . . 15 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
12	11	simprd 113	. . . . . . . . . . . . . 14 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. )
13		vex 2658	. . . . . . . . . . . . . . . . 17 |- u e. _V
14		vex 2658	. . . . . . . . . . . . . . . . 17 |- f e. _V
15	13, 14	opth2 4120	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. <-> ( ( 1st ` y ) = u /\ ( 2nd ` y ) = f ) )
16	15	simplbi 270	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. -> ( 1st ` y ) = u )
17	16	eleq1d 2181	. . . . . . . . . . . . . 14 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. -> ( ( 1st ` y ) e. N. <-> u e. N. ) )
18	12, 17	syl 14	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( ( 1st ` y ) e. N. <-> u e. N. ) )
19	10, 18	mpbid 146	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> u e. N. )
20		xp2nd 6015	. . . . . . . . . . . . . 14 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
21	20	ad2antrr 477	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 2nd ` x ) e. N. )
22	11	simpld 111	. . . . . . . . . . . . . 14 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. )
23		vex 2658	. . . . . . . . . . . . . . . . 17 |- w e. _V
24		vex 2658	. . . . . . . . . . . . . . . . 17 |- v e. _V
25	23, 24	opth2 4120	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. <-> ( ( 1st ` x ) = w /\ ( 2nd ` x ) = v ) )
26	25	simprbi 271	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. -> ( 2nd ` x ) = v )
27	26	eleq1d 2181	. . . . . . . . . . . . . 14 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. -> ( ( 2nd ` x ) e. N. <-> v e. N. ) )
28	22, 27	syl 14	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( ( 2nd ` x ) e. N. <-> v e. N. ) )
29	21, 28	mpbid 146	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> v e. N. )
30		mulcompig 7080	. . . . . . . . . . . 12 |- ( ( u e. N. /\ v e. N. ) -> ( u .N v ) = ( v .N u ) )
31	19, 29, 30	syl2anc 406	. . . . . . . . . . 11 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( u .N v ) = ( v .N u ) )
32	31	oveq2d 5742	. . . . . . . . . 10 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( ( w .N f ) +N ( u .N v ) ) = ( ( w .N f ) +N ( v .N u ) ) )
33	32	opeq1d 3675	. . . . . . . . 9 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. )
34	33	eqeq2d 2124	. . . . . . . 8 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. <-> z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) )
35	34	pm5.32da 445	. . . . . . 7 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) <-> ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) )
36	8, 35	bitr3d 189	. . . . . 6 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) <-> ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) )
37	36	4exbidv 1822	. . . . 5 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( E. w E. v E. u E. f ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) <-> E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) )
38		xp1st 6014	. . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
39	38, 20	jca 302	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) )
40		xp2nd 6015	. . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
41	9, 40	jca 302	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) )
42		simpll 501	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
43		simprr 504	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
44	42, 43	oveq12d 5744	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N f ) = ( ( 1st ` x ) .N ( 2nd ` y ) ) )
45		simprl 503	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
46		simplr 502	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
47	45, 46	oveq12d 5744	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( u .N v ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) )
48	44, 47	oveq12d 5744	. . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( ( w .N f ) +N ( u .N v ) ) = ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) )
49	46, 43	oveq12d 5744	. . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( v .N f ) = ( ( 2nd ` x ) .N ( 2nd ` y ) ) )
50	48, 49	opeq12d 3677	. . . . . . . 8 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
51	50	eqeq2d 2124	. . . . . . 7 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. <-> z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
52	51	copsex4g 4127	. . . . . 6 |- ( ( ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) /\ ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) ) -> ( E. w E. v E. u E. f ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) <-> z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
53	39, 41, 52	syl2an 285	. . . . 5 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( E. w E. v E. u E. f ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( u .N v ) ) , ( v .N f ) >. ) <-> z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
54	37, 53	bitr3d 189	. . . 4 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) <-> z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
55	54	pm5.32i 447	. . 3 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) <-> ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
56	55	oprabbii 5778	. 2 |- { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ z = <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) }
57	1, 2, 56	3eqtr4i 2143	1 |- +pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) }

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1312   E.wex 1449   e. wcel 1461   <.cop 3494   X. cxp 4495   ` cfv 5079  (class class class)co 5726   {coprab 5727   e. cmpo 5728   1stc1st 5987   2ndc2nd 5988   N.cnpi 7021   +N cpli 7022   .N cmi 7023   +pQ cplpq 7025

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5989  df-2nd 5990  df-recs 6153  df-irdg 6218  df-oadd 6268  df-omul 6269  df-ni 7053  df-mi 7055  df-plpq 7093

This theorem is referenced by:  addpipqqs  7119