Theorem soinxp 4333

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 4332	. . 3 ⊢ (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)
2		brinxp 4331	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A) → (x𝑅y ↔ x(𝑅 ∩ (A × A))y))
3	2	3adant3 910	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅y ↔ x(𝑅 ∩ (A × A))y))
4		brinxp 4331	. . . . . . . . 9 ⊢ ((x ∈ A ∧ z ∈ A) → (x𝑅z ↔ x(𝑅 ∩ (A × A))z))
5	4	3adant2 909	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅z ↔ x(𝑅 ∩ (A × A))z))
6		brinxp 4331	. . . . . . . . . 10 ⊢ ((z ∈ A ∧ y ∈ A) → (z𝑅y ↔ z(𝑅 ∩ (A × A))y))
7	6	ancoms 255	. . . . . . . . 9 ⊢ ((y ∈ A ∧ z ∈ A) → (z𝑅y ↔ z(𝑅 ∩ (A × A))y))
8	7	3adant1 908	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (z𝑅y ↔ z(𝑅 ∩ (A × A))y))
9	5, 8	orbi12d 694	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ((x𝑅z ∨ z𝑅y) ↔ (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y)))
10	3, 9	imbi12d 223	. . . . . 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 1089	. . . . 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 2320	. . . 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 2312	. . 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 436	. 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 4004	. 2 ⊢ (𝑅 Or A ↔ (𝑅 Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y))))
16		df-iso 4004	. 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 201	1 ⊢ (𝑅 Or A ↔ (𝑅 ∩ (A × A)) Or A)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   ∧ w3a 871   ∈ wcel 1370  ∀wral 2280   ∩ cin 2889   class class class wbr 3734   Po wpo 4001   Or wor 4002   × cxp 4266

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-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-po 4003  df-iso 4004  df-xp 4274

This theorem is referenced by: (None)