Theorem soinxp 4567

Description: Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
soinxp	⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Proof of Theorem soinxp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poinxp 4566	. . 3 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2		brinxp 4565	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3	2	3adant3 982	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4		brinxp 4565	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
5	4	3adant2 981	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6		brinxp 4565	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧𝑅𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7	6	ancoms 266	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧𝑅𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
8	7	3adant1 980	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧𝑅𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
9	5, 8	orbi12d 765	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
10	3, 9	imbi12d 233	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
11	10	3expb 1163	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
12	11	2ralbidva 2429	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
13	12	ralbiia 2421	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
14	1, 13	anbi12i 453	. 2 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
15		df-iso 4177	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
16		df-iso 4177	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 → (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))))
17	14, 15, 16	3bitr4i 211	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680   ∧ w3a 943   ∈ wcel 1461  ∀wral 2388   ∩ cin 3034   class class class wbr 3893   Po wpo 4174   Or wor 4175   × cxp 4495

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-po 4176  df-iso 4177  df-xp 4503

This theorem is referenced by: (None)