Theorem dfmpq2 7296

Description: Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)

Assertion

Ref	Expression
dfmpq2	⊢ ·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Proof of Theorem dfmpq2

Step	Hyp	Ref	Expression
1		df-mpo 5847	. 2 ⊢ (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)}
2		df-mpq 7286	. 2 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
3		1st2nd2 6143	. . . . . . . . . 10 ⊢ (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
4	3	eqeq1d 2174	. . . . . . . . 9 ⊢ (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5		1st2nd2 6143	. . . . . . . . . 10 ⊢ (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
6	5	eqeq1d 2174	. . . . . . . . 9 ⊢ (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩))
7	4, 6	bi2anan9 596	. . . . . . . 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 461	. . . . . . 7 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)))
9	8	bicomd 140	. . . . . 6 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)))
10	9	4exbidv 1858	. . . . 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 𝑢), (𝑣 ·N 𝑓)⟩) ↔ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)))
11		xp1st 6133	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
12		xp2nd 6134	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
13	11, 12	jca 304	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑥) ∈ N))
14		xp1st 6133	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
15		xp2nd 6134	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
16	14, 15	jca 304	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → ((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑦) ∈ N))
17		simpll 519	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑤 = (1st ‘𝑥))
18		simprl 521	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑢 = (1st ‘𝑦))
19	17, 18	oveq12d 5860	. . . . . . . . 9 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑤 ·N 𝑢) = ((1st ‘𝑥) ·N (1st ‘𝑦)))
20		simplr 520	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑣 = (2nd ‘𝑥))
21		simprr 522	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑓 = (2nd ‘𝑦))
22	20, 21	oveq12d 5860	. . . . . . . . 9 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑣 ·N 𝑓) = ((2nd ‘𝑥) ·N (2nd ‘𝑦)))
23	19, 22	opeq12d 3766	. . . . . . . 8 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
24	23	eqeq2d 2177	. . . . . . 7 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
25	24	copsex4g 4225	. . . . . 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 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
26	13, 16, 25	syl2an 287	. . . . 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 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
27	10, 26	bitr3d 189	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
28	27	pm5.32i 450	. . 3 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩)) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
29	28	oprabbii 5897	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)}
30	1, 2, 29	3eqtr4i 2196	1 ⊢ ·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ⟨cop 3579   × cxp 4602  ‘cfv 5188  (class class class)co 5842  {coprab 5843   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107  Ncnpi 7213   ·_N cmi 7215   ·_pQ cmpq 7218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-mpq 7286

This theorem is referenced by:  mulpipqqs  7314