Theorem ltsosr 7984

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 7983	. 2 ⊢ <R Po R
2		df-nr 7947	. . . 4 ⊢ R = ((P × P) / ~R )
3		breq1 4091	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ 𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4		breq1 4091	. . . . . 6 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ 𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
5	4	orbi1d 798	. . . . 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 234	. . . 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 4092	. . . . 5 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R ↔ 𝑥 <R 𝑦))
8		breq2 4092	. . . . . 6 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
9	8	orbi2d 797	. . . . 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 234	. . . 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 4092	. . . . . 6 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ↔ 𝑥 <R 𝑧))
12		breq1 4091	. . . . . 6 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦 ↔ 𝑧 <R 𝑦))
13	11, 12	orbi12d 800	. . . . 5 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦) ↔ (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
14	13	imbi2d 230	. . . 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 1047	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑎 ∈ P)
16		simp3r 1052	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑓 ∈ P)
17		addclpr 7757	. . . . . . . . 9 ⊢ ((𝑎 ∈ P ∧ 𝑓 ∈ P) → (𝑎 +P 𝑓) ∈ P)
18	15, 16, 17	syl2anc 411	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑎 +P 𝑓) ∈ P)
19		simp2r 1050	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑑 ∈ P)
20		addclpr 7757	. . . . . . . 8 ⊢ (((𝑎 +P 𝑓) ∈ P ∧ 𝑑 ∈ P) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
21	18, 19, 20	syl2anc 411	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
22		simp2l 1049	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑐 ∈ P)
23		addclpr 7757	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑐 ∈ P) → (𝑓 +P 𝑐) ∈ P)
24	16, 22, 23	syl2anc 411	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑓 +P 𝑐) ∈ P)
25		simp1r 1048	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑏 ∈ P)
26		addclpr 7757	. . . . . . . 8 ⊢ (((𝑓 +P 𝑐) ∈ P ∧ 𝑏 ∈ P) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
27	24, 25, 26	syl2anc 411	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
28		simp3l 1051	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑒 ∈ P)
29		addclpr 7757	. . . . . . . . 9 ⊢ ((𝑏 ∈ P ∧ 𝑒 ∈ P) → (𝑏 +P 𝑒) ∈ P)
30	25, 28, 29	syl2anc 411	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑏 +P 𝑒) ∈ P)
31		addclpr 7757	. . . . . . . 8 ⊢ (((𝑏 +P 𝑒) ∈ P ∧ 𝑑 ∈ P) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
32	30, 19, 31	syl2anc 411	. . . . . . 7 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
33		ltsopr 7816	. . . . . . . 8 ⊢ <P Or P
34		sowlin 4417	. . . . . . . 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 424	. . . . . . 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 1273	. . . . . 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 7757	. . . . . . . . 9 ⊢ ((𝑎 ∈ P ∧ 𝑑 ∈ P) → (𝑎 +P 𝑑) ∈ P)
38	15, 19, 37	syl2anc 411	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑎 +P 𝑑) ∈ P)
39		addclpr 7757	. . . . . . . . 9 ⊢ ((𝑏 ∈ P ∧ 𝑐 ∈ P) → (𝑏 +P 𝑐) ∈ P)
40	25, 22, 39	syl2anc 411	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑏 +P 𝑐) ∈ P)
41		ltaprg 7839	. . . . . . . 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 1273	. . . . . . 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 7798	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
44	43	adantl 277	. . . . . . . . . 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 7799	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
46	45	adantl 277	. . . . . . . . . 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 6204	. . . . . . . . 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 6182	. . . . . . . . 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 2267	. . . . . . . 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 6182	. . . . . . . . 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 6203	. . . . . . . . 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 2266	. . . . . . . 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 4101	. . . . . . 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 188	. . . . . 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 7839	. . . . . . . . 9 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
56	55	adantl 277	. . . . . . . 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 6192	. . . . . . 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 7757	. . . . . . . . . 10 ⊢ ((𝑒 ∈ P ∧ 𝑑 ∈ P) → (𝑒 +P 𝑑) ∈ P)
59	28, 19, 58	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → (𝑒 +P 𝑑) ∈ P)
60	56, 59, 24, 25, 44	caovord2d 6192	. . . . . . . 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 6182	. . . . . . . . . 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 6179	. . . . . . . . . 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 2264	. . . . . . . . 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 4098	. . . . . . . 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 191	. . . . . . 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 800	. . . . . 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 203	. . . . 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 7967	. . . . . 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 1043	. . . . 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 7967	. . . . . . 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 1042	. . . . . 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 7967	. . . . . . . 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 268	. . . . . . 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 1041	. . . . . 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 800	. . . . 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 203	. . . 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 6793	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
78	77	rgen3 2619	. 2 ⊢ ∀𝑥 ∈ R ∀𝑦 ∈ R ∀𝑧 ∈ R (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))
79		df-iso 4394	. 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 950	1 ⊢ <R Or R

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ⟨cop 3672   class class class wbr 4088   Po wpo 4391   Or wor 4392  (class class class)co 6018  [cec 6700  Pcnp 7511   +_P cpp 7513  <_P cltp 7515   ~_R cer 7516  Rcnr 7517   <_R cltr 7523

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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-iplp 7688  df-iltp 7690  df-enr 7946  df-nr 7947  df-ltr 7950

This theorem is referenced by:  1ne0sr  7986  addgt0sr  7995  caucvgsrlemcl  8009  caucvgsrlemfv  8011  suplocsrlemb  8026  suplocsrlempr  8027  suplocsrlem  8028  axpre-ltirr  8102  axpre-ltwlin  8103  axpre-lttrn  8104