Theorem ltsosr 7726

Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)

Assertion

Ref	Expression
ltsosr	⊢ <R Or R

Proof of Theorem ltsosr

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltposr 7725	. 2 ⊢ <R Po R
2		df-nr 7689	. . . 4 ⊢ R = ((P × P) / ~R )
3		breq1 3992	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ 𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4		breq1 3992	. . . . . 6 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ 𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
5	4	orbi1d 786	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )))
6	3, 5	imbi12d 233	. . . 4 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )) ↔ (𝑥 <R [⟨𝑐, 𝑑⟩] ~R → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ))))
7		breq2 3993	. . . . 5 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R ↔ 𝑥 <R 𝑦))
8		breq2 3993	. . . . . 6 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
9	8	orbi2d 785	. . . . 5 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦)))
10	7, 9	imbi12d 233	. . . 4 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ((𝑥 <R [⟨𝑐, 𝑑⟩] ~R → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )) ↔ (𝑥 <R 𝑦 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))))
11		breq2 3993	. . . . . 6 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ↔ 𝑥 <R 𝑧))
12		breq1 3992	. . . . . 6 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦 ↔ 𝑧 <R 𝑦))
13	11, 12	orbi12d 788	. . . . 5 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦) ↔ (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
14	13	imbi2d 229	. . . 4 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R 𝑦 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦)) ↔ (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))))
15		simp1l 1016	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑎 ∈ P)
16		simp3r 1021	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑓 ∈ P)
17		addclpr 7499	. . . . . . . . 9 ⊢ ((𝑎 ∈ P ∧ 𝑓 ∈ P) → (𝑎 +P 𝑓) ∈ P)
18	15, 16, 17	syl2anc 409	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑎 +P 𝑓) ∈ P)
19		simp2r 1019	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑑 ∈ P)
20		addclpr 7499	. . . . . . . 8 ⊢ (((𝑎 +P 𝑓) ∈ P ∧ 𝑑 ∈ P) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
21	18, 19, 20	syl2anc 409	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
22		simp2l 1018	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑐 ∈ P)
23		addclpr 7499	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑐 ∈ P) → (𝑓 +P 𝑐) ∈ P)
24	16, 22, 23	syl2anc 409	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑓 +P 𝑐) ∈ P)
25		simp1r 1017	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑏 ∈ P)
26		addclpr 7499	. . . . . . . 8 ⊢ (((𝑓 +P 𝑐) ∈ P ∧ 𝑏 ∈ P) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
27	24, 25, 26	syl2anc 409	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
28		simp3l 1020	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑒 ∈ P)
29		addclpr 7499	. . . . . . . . 9 ⊢ ((𝑏 ∈ P ∧ 𝑒 ∈ P) → (𝑏 +P 𝑒) ∈ P)
30	25, 28, 29	syl2anc 409	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑏 +P 𝑒) ∈ P)
31		addclpr 7499	. . . . . . . 8 ⊢ (((𝑏 +P 𝑒) ∈ P ∧ 𝑑 ∈ P) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
32	30, 19, 31	syl2anc 409	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
33		ltsopr 7558	. . . . . . . 8 ⊢ <P Or P
34		sowlin 4305	. . . . . . . 8 ⊢ ((<P Or P ∧ (((𝑎 +P 𝑓) +P 𝑑) ∈ P ∧ ((𝑓 +P 𝑐) +P 𝑏) ∈ P ∧ ((𝑏 +P 𝑒) +P 𝑑) ∈ P)) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
35	33, 34	mpan 422	. . . . . . 7 ⊢ ((((𝑎 +P 𝑓) +P 𝑑) ∈ P ∧ ((𝑓 +P 𝑐) +P 𝑏) ∈ P ∧ ((𝑏 +P 𝑒) +P 𝑑) ∈ P) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
36	21, 27, 32, 35	syl3anc 1233	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
37		addclpr 7499	. . . . . . . . 9 ⊢ ((𝑎 ∈ P ∧ 𝑑 ∈ P) → (𝑎 +P 𝑑) ∈ P)
38	15, 19, 37	syl2anc 409	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑎 +P 𝑑) ∈ P)
39		addclpr 7499	. . . . . . . . 9 ⊢ ((𝑏 ∈ P ∧ 𝑐 ∈ P) → (𝑏 +P 𝑐) ∈ P)
40	25, 22, 39	syl2anc 409	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑏 +P 𝑐) ∈ P)
41		ltaprg 7581	. . . . . . . 8 ⊢ (((𝑎 +P 𝑑) ∈ P ∧ (𝑏 +P 𝑐) ∈ P ∧ 𝑓 ∈ P) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
42	38, 40, 16, 41	syl3anc 1233	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
43		addcomprg 7540	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
44	43	adantl 275	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) ∧ (𝑟 ∈ P ∧ 𝑠 ∈ P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
45		addassprg 7541	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
46	45	adantl 275	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) ∧ (𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
47	16, 15, 19, 44, 46	caov12d 6034	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑓 +P (𝑎 +P 𝑑)) = (𝑎 +P (𝑓 +P 𝑑)))
48	46, 15, 16, 19	caovassd 6012	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑓) +P 𝑑) = (𝑎 +P (𝑓 +P 𝑑)))
49	47, 48	eqtr4d 2206	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑓 +P (𝑎 +P 𝑑)) = ((𝑎 +P 𝑓) +P 𝑑))
50	46, 16, 25, 22	caovassd 6012	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑓 +P 𝑏) +P 𝑐) = (𝑓 +P (𝑏 +P 𝑐)))
51	16, 25, 22, 44, 46	caov32d 6033	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑓 +P 𝑏) +P 𝑐) = ((𝑓 +P 𝑐) +P 𝑏))
52	50, 51	eqtr3d 2205	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑓 +P (𝑏 +P 𝑐)) = ((𝑓 +P 𝑐) +P 𝑏))
53	49, 52	breq12d 4002	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐)) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
54	42, 53	bitrd 187	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
55		ltaprg 7581	. . . . . . . . 9 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
56	55	adantl 275	. . . . . . . 8 ⊢ ((((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) ∧ (𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
57	56, 18, 30, 19, 44	caovord2d 6022	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑)))
58		addclpr 7499	. . . . . . . . . 10 ⊢ ((𝑒 ∈ P ∧ 𝑑 ∈ P) → (𝑒 +P 𝑑) ∈ P)
59	28, 19, 58	syl2anc 409	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑒 +P 𝑑) ∈ P)
60	56, 59, 24, 25, 44	caovord2d 6022	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑒 +P 𝑑) +P 𝑏)<P ((𝑓 +P 𝑐) +P 𝑏)))
61	46, 25, 28, 19	caovassd 6012	. . . . . . . . . 10 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑏 +P 𝑒) +P 𝑑) = (𝑏 +P (𝑒 +P 𝑑)))
62	44, 25, 59	caovcomd 6009	. . . . . . . . . 10 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑏 +P (𝑒 +P 𝑑)) = ((𝑒 +P 𝑑) +P 𝑏))
63	61, 62	eqtrd 2203	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑏 +P 𝑒) +P 𝑑) = ((𝑒 +P 𝑑) +P 𝑏))
64	63	breq1d 3999	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) ↔ ((𝑒 +P 𝑑) +P 𝑏)<P ((𝑓 +P 𝑐) +P 𝑏)))
65	60, 64	bitr4d 190	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
66	57, 65	orbi12d 788	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)) ↔ (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
67	36, 54, 66	3imtr4d 202	. . . . 5 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐))))
68		ltsrprg 7709	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
69	68	3adant3 1012	. . . . 5 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
70		ltsrprg 7709	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
71	70	3adant2 1011	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
72		ltsrprg 7709	. . . . . . . 8 ⊢ (((𝑒 ∈ P ∧ 𝑓 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
73	72	ancoms 266	. . . . . . 7 ⊢ (((𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
74	73	3adant1 1010	. . . . . 6 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
75	71, 74	orbi12d 788	. . . . 5 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐))))
76	67, 69, 75	3imtr4d 202	. . . 4 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )))
77	2, 6, 10, 14, 76	3ecoptocl 6602	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
78	77	rgen3 2557	. 2 ⊢ ∀𝑥 ∈ R ∀𝑦 ∈ R ∀𝑧 ∈ R (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))
79		df-iso 4282	. 2 ⊢ ( <R Or R ↔ ( <R Po R ∧ ∀𝑥 ∈ R ∀𝑦 ∈ R ∀𝑧 ∈ R (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))))
80	1, 78, 79	mpbir2an 937	1 ⊢ <R Or R

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ⟨cop 3586   class class class wbr 3989   Po wpo 4279   Or wor 4280  (class class class)co 5853  [cec 6511  Pcnp 7253   +_P cpp 7255  <_P cltp 7257   ~_R cer 7258  Rcnr 7259   <_R cltr 7265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432  df-enr 7688  df-nr 7689  df-ltr 7692

This theorem is referenced by:  1ne0sr  7728  addgt0sr  7737  caucvgsrlemcl  7751  caucvgsrlemfv  7753  suplocsrlemb  7768  suplocsrlempr  7769  suplocsrlem  7770  axpre-ltirr  7844  axpre-ltwlin  7845  axpre-lttrn  7846