Theorem isprs 18367

Description: Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypotheses

Ref	Expression
isprs.b	⊢ 𝐵 = (Base‘𝐾)
isprs.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
isprs	⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥, ≤ ,𝑦,𝑧

Proof of Theorem isprs

Dummy variables 𝑓 𝑏 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6920	. . . 4 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2		fveq2 6920	. . . . 5 ⊢ (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
3	2	sbceq1d 3809	. . . 4 ⊢ (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ [(le‘𝐾) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
4	1, 3	sbceqbid 3811	. . 3 ⊢ (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
5		fvex 6933	. . . 4 ⊢ (Base‘𝐾) ∈ V
6		fvex 6933	. . . 4 ⊢ (le‘𝐾) ∈ V
7		isprs.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		eqtr3 2766	. . . . . . 7 ⊢ ((𝑏 = (Base‘𝐾) ∧ 𝐵 = (Base‘𝐾)) → 𝑏 = 𝐵)
9	7, 8	mpan2 690	. . . . . 6 ⊢ (𝑏 = (Base‘𝐾) → 𝑏 = 𝐵)
10		raleq 3331	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
11	10	raleqbi1dv 3346	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
12	11	raleqbi1dv 3346	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
13	9, 12	syl 17	. . . . 5 ⊢ (𝑏 = (Base‘𝐾) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
14		isprs.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
15		eqtr3 2766	. . . . . . 7 ⊢ ((𝑟 = (le‘𝐾) ∧ ≤ = (le‘𝐾)) → 𝑟 = ≤ )
16	14, 15	mpan2 690	. . . . . 6 ⊢ (𝑟 = (le‘𝐾) → 𝑟 = ≤ )
17		breq 5168	. . . . . . . . 9 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑥 ↔ 𝑥 ≤ 𝑥))
18		breq 5168	. . . . . . . . . . 11 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑦 ↔ 𝑥 ≤ 𝑦))
19		breq 5168	. . . . . . . . . . 11 ⊢ (𝑟 = ≤ → (𝑦𝑟𝑧 ↔ 𝑦 ≤ 𝑧))
20	18, 19	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧)))
21		breq 5168	. . . . . . . . . 10 ⊢ (𝑟 = ≤ → (𝑥𝑟𝑧 ↔ 𝑥 ≤ 𝑧))
22	20, 21	imbi12d 344	. . . . . . . . 9 ⊢ (𝑟 = ≤ → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
23	17, 22	anbi12d 631	. . . . . . . 8 ⊢ (𝑟 = ≤ → ((𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
24	23	ralbidv 3184	. . . . . . 7 ⊢ (𝑟 = ≤ → (∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
25	24	2ralbidv 3227	. . . . . 6 ⊢ (𝑟 = ≤ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
26	16, 25	syl 17	. . . . 5 ⊢ (𝑟 = (le‘𝐾) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
27	13, 26	sylan9bb 509	. . . 4 ⊢ ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
28	5, 6, 27	sbc2ie 3887	. . 3 ⊢ ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
29	4, 28	bitrdi 287	. 2 ⊢ (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
30		df-proset 18365	. 2 ⊢ Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
31	29, 30	elab4g 3699	1 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  [wsbc 3804   class class class wbr 5166  ‘cfv 6573  Basecbs 17258  lecple 17318   Proset cproset 18363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-proset 18365

This theorem is referenced by:  prslem  18368  ispos2  18385  ressprs  32936  oduprs  32937  isprsd  48635