Theorem dfplpq2 6544

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-mpt2 5537	. 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 6534	. 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 5821	. . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
4	3	eqeq1d 2089	. . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5		1st2nd2 5821	. . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
6	5	eqeq1d 2089	. . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
7	4, 6	bi2anan9 570	. . . . . . . 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 452	. . . . . . 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 5812	. . . . . . . . . . . . . 14 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
10	9	ad2antlr 472	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 1st ` y ) e. N. )
11	7	biimpa 290	. . . . . . . . . . . . . . 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 112	. . . . . . . . . . . . . 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 2604	. . . . . . . . . . . . . . . . 17 |- u e. _V
14		vex 2604	. . . . . . . . . . . . . . . . 17 |- f e. _V
15	13, 14	opth2 3995	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. <-> ( ( 1st ` y ) = u /\ ( 2nd ` y ) = f ) )
16	15	simplbi 268	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. -> ( 1st ` y ) = u )
17	16	eleq1d 2147	. . . . . . . . . . . . . 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 145	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> u e. N. )
20		xp2nd 5813	. . . . . . . . . . . . . 14 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
21	20	ad2antrr 471	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 2nd ` x ) e. N. )
22	11	simpld 110	. . . . . . . . . . . . . 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 2604	. . . . . . . . . . . . . . . . 17 |- w e. _V
24		vex 2604	. . . . . . . . . . . . . . . . 17 |- v e. _V
25	23, 24	opth2 3995	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. <-> ( ( 1st ` x ) = w /\ ( 2nd ` x ) = v ) )
26	25	simprbi 269	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. -> ( 2nd ` x ) = v )
27	26	eleq1d 2147	. . . . . . . . . . . . . 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 145	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> v e. N. )
30		mulcompig 6521	. . . . . . . . . . . 12 |- ( ( u e. N. /\ v e. N. ) -> ( u .N v ) = ( v .N u ) )
31	19, 29, 30	syl2anc 403	. . . . . . . . . . 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 5548	. . . . . . . . . 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 3576	. . . . . . . . 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 2092	. . . . . . . 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 439	. . . . . . 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 188	. . . . . 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 1791	. . . . 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 5812	. . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
39	38, 20	jca 300	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) )
40		xp2nd 5813	. . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
41	9, 40	jca 300	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) )
42		simpll 495	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
43		simprr 498	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
44	42, 43	oveq12d 5550	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N f ) = ( ( 1st ` x ) .N ( 2nd ` y ) ) )
45		simprl 497	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
46		simplr 496	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
47	45, 46	oveq12d 5550	. . . . . . . . . 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 5550	. . . . . . . . 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 5550	. . . . . . . . 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 3578	. . . . . . . 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 2092	. . . . . . 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 4002	. . . . . 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 283	. . . . 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 188	. . . 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 441	. . 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 5580	. 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 2111	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 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   <.cop 3401   X. cxp 4361   ` cfv 4922  (class class class)co 5532   {coprab 5533   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786   N.cnpi 6462   +N cpli 6463   .N cmi 6464   +pQ cplpq 6466

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-ni 6494  df-mi 6496  df-plpq 6534

This theorem is referenced by:  addpipqqs  6560