Theorem dfmpq2 6453

Description: Alternative 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-mpt2 5517	. 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 6443	. 2 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
3		1st2nd2 5801	. . . . . . . . . 10 ⊢ (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
4	3	eqeq1d 2048	. . . . . . . . 9 ⊢ (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5		1st2nd2 5801	. . . . . . . . . 10 ⊢ (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
6	5	eqeq1d 2048	. . . . . . . . 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 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 129	. . . . . 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 1750	. . . . 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 5792	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
12		xp2nd 5793	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
13	11, 12	jca 290	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑥) ∈ N))
14		xp1st 5792	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
15		xp2nd 5793	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
16	14, 15	jca 290	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → ((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑦) ∈ N))
17		simpll 481	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑤 = (1st ‘𝑥))
18		simprl 483	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑢 = (1st ‘𝑦))
19	17, 18	oveq12d 5530	. . . . . . . . 9 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑤 ·N 𝑢) = ((1st ‘𝑥) ·N (1st ‘𝑦)))
20		simplr 482	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑣 = (2nd ‘𝑥))
21		simprr 484	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑓 = (2nd ‘𝑦))
22	20, 21	oveq12d 5530	. . . . . . . . 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 3557	. . . . . . . 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 2051	. . . . . . 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 3984	. . . . . 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 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 𝑢), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
27	10, 26	bitr3d 179	. . . 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 427	. . 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 5560	. 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 2070	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 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   × cxp 4343  ‘cfv 4902  (class class class)co 5512  {coprab 5513   ↦ cmpt2 5514  1^st c1st 5765  2^nd c2nd 5766  Ncnpi 6370   ·_N cmi 6372   ·_pQ cmpq 6375

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-mpq 6443

This theorem is referenced by:  mulpipqqs  6471