Theorem dfmpq2 7317

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 5858	. 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 7307	. 2 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
3		1st2nd2 6154	. . . . . . . . . 10 ⊢ (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
4	3	eqeq1d 2179	. . . . . . . . 9 ⊢ (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5		1st2nd2 6154	. . . . . . . . . 10 ⊢ (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
6	5	eqeq1d 2179	. . . . . . . . 9 ⊢ (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩))
7	4, 6	bi2anan9 601	. . . . . . . 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 462	. . . . . . 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 1863	. . . . 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 6144	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
12		xp2nd 6145	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
13	11, 12	jca 304	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑥) ∈ N))
14		xp1st 6144	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
15		xp2nd 6145	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
16	14, 15	jca 304	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → ((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑦) ∈ N))
17		simpll 524	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑤 = (1st ‘𝑥))
18		simprl 526	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑢 = (1st ‘𝑦))
19	17, 18	oveq12d 5871	. . . . . . . . 9 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑤 ·N 𝑢) = ((1st ‘𝑥) ·N (1st ‘𝑦)))
20		simplr 525	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑣 = (2nd ‘𝑥))
21		simprr 527	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑓 = (2nd ‘𝑦))
22	20, 21	oveq12d 5871	. . . . . . . . 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 3773	. . . . . . . 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 2182	. . . . . . 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 4232	. . . . . 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 451	. . 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 5908	. 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 2201	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 1348  ∃wex 1485   ∈ wcel 2141  ⟨cop 3586   × cxp 4609  ‘cfv 5198  (class class class)co 5853  {coprab 5854   ∈ cmpo 5855  1^st c1st 6117  2^nd c2nd 6118  Ncnpi 7234   ·_N cmi 7236   ·_pQ cmpq 7239

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-mpq 7307

This theorem is referenced by:  mulpipqqs  7335