Theorem dfplpq2 6338

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.wE.vE.uE.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 5460	. 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 6328	. 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 5743	. . . . . . . . . 10 |- (x e. ( N. X. N.) -> x = <.( 1st ` x) , ( 2nd ` x) >.)
4	3	eqeq1d 2045	. . . . . . . . 9 |- (x e. ( N. X. N.) -> (x = <.w , v >. <-> <.( 1st ` x) , ( 2nd ` x) >. = <.w , v >.))
5		1st2nd2 5743	. . . . . . . . . 10 |- (y e. ( N. X. N.) -> y = <.( 1st ` y) , ( 2nd ` y) >.)
6	5	eqeq1d 2045	. . . . . . . . 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 5734	. . . . . . . . . . . . . 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 2554	. . . . . . . . . . . . . . . . 17 |- u e. _V
14		vex 2554	. . . . . . . . . . . . . . . . 17 |- f e. _V
15	13, 14	opth2 3968	. . . . . . . . . . . . . . . 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 2103	. . . . . . . . . . . . . 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 5735	. . . . . . . . . . . . . 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 2554	. . . . . . . . . . . . . . . . 17 |- w e. _V
24		vex 2554	. . . . . . . . . . . . . . . . 17 |- v e. _V
25	23, 24	opth2 3968	. . . . . . . . . . . . . . . 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 2103	. . . . . . . . . . . . . 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 6315	. . . . . . . . . . . 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 5471	. . . . . . . . . 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 3546	. . . . . . . . 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 2048	. . . . . . . 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 1747	. . . . 5 |- ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) -> (E.wE.vE.uE.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.wE.vE.uE.f((x = <.w , v >. /\ y = <.u , f >.) /\ z = <.((w .N f) +N (v .N u)) , (v .N f) >.)))
38		xp1st 5734	. . . . . . 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 5735	. . . . . . 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 5473	. . . . . . . . . 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 5473	. . . . . . . . . 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 5473	. . . . . . . . 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 5473	. . . . . . . . 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 3548	. . . . . . . 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 2048	. . . . . . 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 3975	. . . . . 6 |- (((( 1st ` x) e. N. /\ ( 2nd ` x) e. N.) /\ (( 1st ` y) e. N. /\ ( 2nd ` y) e. N.)) -> (E.wE.vE.uE.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.wE.vE.uE.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.wE.vE.uE.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.wE.vE.uE.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 5502	. 2 |- { <. <.x , y >. , z >. | ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) /\ E.wE.vE.uE.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 2067	1 |- +pQ = { <. <.x , y >. , z >. | ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) /\ E.wE.vE.uE.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 1242  E.wex 1378   e. wcel 1390   <.cop 3370   X. cxp 4286   ` cfv 4845  (class class class)co 5455   {coprab 5456   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   N.cnpi 6256   +N cpli 6257   .N cmi 6258   +pQ cplpq 6260

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-ni 6288  df-mi 6290  df-plpq 6328

This theorem is referenced by:  addpipqqs  6354