Theorem soinxp 4798

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 4797	. . 3 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2		brinxp 4796	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3	2	3adant3 1043	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
4		brinxp 4796	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
5	4	3adant2 1042	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
6		brinxp 4796	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧𝑅𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7	6	ancoms 268	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧𝑅𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
8	7	3adant1 1041	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑧𝑅𝑦 ↔ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦))
9	5, 8	orbi12d 800	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧 ∨ 𝑧(𝑅 ∩ (𝐴 × 𝐴))𝑦)))
10	3, 9	imbi12d 234	. . . . . 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 1230	. . . . 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 2553	. . . 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 2545	. . 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 460	. 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 4396	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
16		df-iso 4396	. 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 212	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   ∧ w3a 1004   ∈ wcel 2201  ∀wral 2509   ∩ cin 3198   class class class wbr 4089   Po wpo 4393   Or wor 4394   × cxp 4725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-po 4395  df-iso 4396  df-xp 4733

This theorem is referenced by: (None)