Theorem dfplpq2 6452

Description: Alternative 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 5517	. 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 6442	. 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 5801	. . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
4	3	eqeq1d 2048	. . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5		1st2nd2 5801	. . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
6	5	eqeq1d 2048	. . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
7	4, 6	bi2anan9 538	. . . . . . . 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 438	. . . . . . 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 5792	. . . . . . . . . . . . . 14 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
10	9	ad2antlr 458	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 1st ` y ) e. N. )
11	7	biimpa 280	. . . . . . . . . . . . . . 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 107	. . . . . . . . . . . . . 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 2560	. . . . . . . . . . . . . . . . 17 |- u e. _V
14		vex 2560	. . . . . . . . . . . . . . . . 17 |- f e. _V
15	13, 14	opth2 3977	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. <-> ( ( 1st ` y ) = u /\ ( 2nd ` y ) = f ) )
16	15	simplbi 259	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. -> ( 1st ` y ) = u )
17	16	eleq1d 2106	. . . . . . . . . . . . . 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 135	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> u e. N. )
20		xp2nd 5793	. . . . . . . . . . . . . 14 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
21	20	ad2antrr 457	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 2nd ` x ) e. N. )
22	11	simpld 105	. . . . . . . . . . . . . 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 2560	. . . . . . . . . . . . . . . . 17 |- w e. _V
24		vex 2560	. . . . . . . . . . . . . . . . 17 |- v e. _V
25	23, 24	opth2 3977	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. <-> ( ( 1st ` x ) = w /\ ( 2nd ` x ) = v ) )
26	25	simprbi 260	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. -> ( 2nd ` x ) = v )
27	26	eleq1d 2106	. . . . . . . . . . . . . 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 135	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> v e. N. )
30		mulcompig 6429	. . . . . . . . . . . 12 |- ( ( u e. N. /\ v e. N. ) -> ( u .N v ) = ( v .N u ) )
31	19, 29, 30	syl2anc 391	. . . . . . . . . . 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 5528	. . . . . . . . . 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 3555	. . . . . . . . 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 2051	. . . . . . . 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 425	. . . . . . 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 179	. . . . . 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 1750	. . . . 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 5792	. . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
39	38, 20	jca 290	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) )
40		xp2nd 5793	. . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
41	9, 40	jca 290	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) )
42		simpll 481	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
43		simprr 484	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
44	42, 43	oveq12d 5530	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N f ) = ( ( 1st ` x ) .N ( 2nd ` y ) ) )
45		simprl 483	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
46		simplr 482	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
47	45, 46	oveq12d 5530	. . . . . . . . . 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 5530	. . . . . . . . 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 5530	. . . . . . . . 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 3557	. . . . . . . 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 2051	. . . . . . 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 3984	. . . . . 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 273	. . . . 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 179	. . . 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 427	. . 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 5560	. 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 2070	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 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378   X. cxp 4343   ` cfv 4902  (class class class)co 5512   {coprab 5513   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   N.cnpi 6370   +N cpli 6371   .N cmi 6372   +pQ cplpq 6374

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-ni 6402  df-mi 6404  df-plpq 6442

This theorem is referenced by:  addpipqqs  6468