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 ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃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 ∈ (N × N), y ∈ (N × N) ↦ ⟨(((1st ‘x) ·N (2nd ‘y)) +N ((1st ‘y) ·N (2nd ‘x))), ((2nd ‘x) ·N (2nd ‘y))⟩) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N), y ∈ (N × N) ↦ ⟨(((1st ‘x) ·N (2nd ‘y)) +N ((1st ‘y) ·N (2nd ‘x))), ((2nd ‘x) ·N (2nd ‘y))⟩)
3		1st2nd2 5743	. . . . . . . . . 10 ⊢ (x ∈ (N × N) → x = ⟨(1st ‘x), (2nd ‘x)⟩)
4	3	eqeq1d 2045	. . . . . . . . 9 ⊢ (x ∈ (N × N) → (x = ⟨w, v⟩ ↔ ⟨(1st ‘x), (2nd ‘x)⟩ = ⟨w, v⟩))
5		1st2nd2 5743	. . . . . . . . . 10 ⊢ (y ∈ (N × N) → y = ⟨(1st ‘y), (2nd ‘y)⟩)
6	5	eqeq1d 2045	. . . . . . . . 9 ⊢ (y ∈ (N × N) → (y = ⟨u, f⟩ ↔ ⟨(1st ‘y), (2nd ‘y)⟩ = ⟨u, f⟩))
7	4, 6	bi2anan9 538	. . . . . . . 8 ⊢ ((x ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) → (1st ‘y) ∈ N)
10	9	ad2antlr 458	. . . . . . . . . . . . 13 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → (1st ‘y) ∈ N)
11	7	biimpa 280	. . . . . . . . . . . . . . 15 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → ⟨(1st ‘y), (2nd ‘y)⟩ = ⟨u, f⟩)
13		vex 2554	. . . . . . . . . . . . . . . . 17 ⊢ u ∈ V
14		vex 2554	. . . . . . . . . . . . . . . . 17 ⊢ f ∈ 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) ∈ N ↔ u ∈ N))
18	12, 17	syl 14	. . . . . . . . . . . . 13 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → ((1st ‘y) ∈ N ↔ u ∈ N))
19	10, 18	mpbid 135	. . . . . . . . . . . 12 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → u ∈ N)
20		xp2nd 5735	. . . . . . . . . . . . . 14 ⊢ (x ∈ (N × N) → (2nd ‘x) ∈ N)
21	20	ad2antrr 457	. . . . . . . . . . . . 13 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → (2nd ‘x) ∈ N)
22	11	simpld 105	. . . . . . . . . . . . . 14 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → ⟨(1st ‘x), (2nd ‘x)⟩ = ⟨w, v⟩)
23		vex 2554	. . . . . . . . . . . . . . . . 17 ⊢ w ∈ V
24		vex 2554	. . . . . . . . . . . . . . . . 17 ⊢ v ∈ 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) ∈ N ↔ v ∈ N))
28	22, 27	syl 14	. . . . . . . . . . . . 13 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → ((2nd ‘x) ∈ N ↔ v ∈ N))
29	21, 28	mpbid 135	. . . . . . . . . . . 12 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → v ∈ N)
30		mulcompig 6315	. . . . . . . . . . . 12 ⊢ ((u ∈ N ∧ v ∈ N) → (u ·N v) = (v ·N u))
31	19, 29, 30	syl2anc 391	. . . . . . . . . . 11 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ (x = ⟨w, v⟩ ∧ y = ⟨u, f⟩)) → (u ·N v) = (v ·N u))
32	31	oveq2d 5471	. . . . . . . . . 10 ⊢ (((x ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × N)) → (∃w∃v∃u∃f((⟨(1st ‘x), (2nd ‘x)⟩ = ⟨w, v⟩ ∧ ⟨(1st ‘y), (2nd ‘y)⟩ = ⟨u, f⟩) ∧ z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)))
38		xp1st 5734	. . . . . . 7 ⊢ (x ∈ (N × N) → (1st ‘x) ∈ N)
39	38, 20	jca 290	. . . . . 6 ⊢ (x ∈ (N × N) → ((1st ‘x) ∈ N ∧ (2nd ‘x) ∈ N))
40		xp2nd 5735	. . . . . . 7 ⊢ (y ∈ (N × N) → (2nd ‘y) ∈ N)
41	9, 40	jca 290	. . . . . 6 ⊢ (y ∈ (N × N) → ((1st ‘y) ∈ N ∧ (2nd ‘y) ∈ 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) ∈ N ∧ (2nd ‘x) ∈ N) ∧ ((1st ‘y) ∈ N ∧ (2nd ‘y) ∈ N)) → (∃w∃v∃u∃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 ∈ (N × N) ∧ y ∈ (N × N)) → (∃w∃v∃u∃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 ∈ (N × N) ∧ y ∈ (N × N)) → (∃w∃v∃u∃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 ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)) ↔ ((x ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ (N × N) ∧ y ∈ (N × 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 ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃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  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370   × cxp 4286  ‘cfv 4845  (class class class)co 5455  {coprab 5456   ↦ cmpt2 5457  1^st c1st 5707  2^nd c2nd 5708  Ncnpi 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