Theorem dfplpq2 7344

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 5874	. 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 7334	. 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 6170	. . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
4	3	eqeq1d 2186	. . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5		1st2nd2 6170	. . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
6	5	eqeq1d 2186	. . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
7	4, 6	bi2anan9 606	. . . . . . . 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 465	. . . . . . 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 6160	. . . . . . . . . . . . . 14 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
10	9	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 1st ` y ) e. N. )
11	7	biimpa 296	. . . . . . . . . . . . . . 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 114	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . 17 |- u e. _V
14		vex 2740	. . . . . . . . . . . . . . . . 17 |- f e. _V
15	13, 14	opth2 4237	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. <-> ( ( 1st ` y ) = u /\ ( 2nd ` y ) = f ) )
16	15	simplbi 274	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. -> ( 1st ` y ) = u )
17	16	eleq1d 2246	. . . . . . . . . . . . . 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 147	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> u e. N. )
20		xp2nd 6161	. . . . . . . . . . . . . 14 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
21	20	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> ( 2nd ` x ) e. N. )
22	11	simpld 112	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . 17 |- w e. _V
24		vex 2740	. . . . . . . . . . . . . . . . 17 |- v e. _V
25	23, 24	opth2 4237	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. <-> ( ( 1st ` x ) = w /\ ( 2nd ` x ) = v ) )
26	25	simprbi 275	. . . . . . . . . . . . . . 15 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. -> ( 2nd ` x ) = v )
27	26	eleq1d 2246	. . . . . . . . . . . . . 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 147	. . . . . . . . . . . 12 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( x = <. w , v >. /\ y = <. u , f >. ) ) -> v e. N. )
30		mulcompig 7321	. . . . . . . . . . . 12 |- ( ( u e. N. /\ v e. N. ) -> ( u .N v ) = ( v .N u ) )
31	19, 29, 30	syl2anc 411	. . . . . . . . . . 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 5885	. . . . . . . . . 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 3782	. . . . . . . . 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 2189	. . . . . . . 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 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 ) >. ) <-> ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) )
36	8, 35	bitr3d 190	. . . . . 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 1870	. . . . 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 6160	. . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
39	38, 20	jca 306	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) )
40		xp2nd 6161	. . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
41	9, 40	jca 306	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) )
42		simpll 527	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
43		simprr 531	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
44	42, 43	oveq12d 5887	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N f ) = ( ( 1st ` x ) .N ( 2nd ` y ) ) )
45		simprl 529	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
46		simplr 528	. . . . . . . . . . 11 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
47	45, 46	oveq12d 5887	. . . . . . . . . 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 5887	. . . . . . . . 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 5887	. . . . . . . . 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 3784	. . . . . . . 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 2189	. . . . . . 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 4244	. . . . . 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 289	. . . . 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 190	. . . 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 454	. . 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 5924	. 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 2208	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 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   <.cop 3594   X. cxp 4621   ` cfv 5212  (class class class)co 5869   {coprab 5870   e. cmpo 5871   1stc1st 6133   2ndc2nd 6134   N.cnpi 7262   +N cpli 7263   .N cmi 7264   +pQ cplpq 7266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416  df-ni 7294  df-mi 7296  df-plpq 7334

This theorem is referenced by:  addpipqqs  7360