Theorem ltsosr 8075

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 8074	. 2 ⊢ <R Po R
2		df-nr 8038	. . . 4 ⊢ R = ((P × P) / ~R )
3		breq1 4111	. . . . 5 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ 𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4		breq1 4111	. . . . . 6 ⊢ ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ 𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
5	4	orbi1d 799	. . . . 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 4112	. . . . 5 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R ↔ 𝑥 <R 𝑦))
8		breq2 4112	. . . . . 6 ⊢ ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
9	8	orbi2d 798	. . . . 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 4112	. . . . . 6 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ↔ 𝑥 <R 𝑧))
12		breq1 4111	. . . . . 6 ⊢ ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦 ↔ 𝑧 <R 𝑦))
13	11, 12	orbi12d 801	. . . . 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 1048	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑎 ∈ P)
16		simp3r 1053	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑓 ∈ P)
17		addclpr 7848	. . . . . . . . 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 1051	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑑 ∈ P)
20		addclpr 7848	. . . . . . . 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 1050	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑐 ∈ P)
23		addclpr 7848	. . . . . . . . 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 1049	. . . . . . . 8 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑏 ∈ P)
26		addclpr 7848	. . . . . . . 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 1052	. . . . . . . . 9 ⊢ (((𝑎 ∈ P ∧ 𝑏 ∈ P) ∧ (𝑐 ∈ P ∧ 𝑑 ∈ P) ∧ (𝑒 ∈ P ∧ 𝑓 ∈ P)) → 𝑒 ∈ P)
29		addclpr 7848	. . . . . . . . 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 7848	. . . . . . . 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 7907	. . . . . . . 8 ⊢ <P Or P
34		sowlin 4440	. . . . . . . 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 1274	. . . . . 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 7848	. . . . . . . . 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 7848	. . . . . . . . 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 7930	. . . . . . . 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 1274	. . . . . . 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 7889	. . . . . . . . . . 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 7890	. . . . . . . . . . 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 6235	. . . . . . . . 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 6213	. . . . . . . . 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 2268	. . . . . . . 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 6213	. . . . . . . . 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 6234	. . . . . . . . 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 2267	. . . . . . . 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 4121	. . . . . . 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 7930	. . . . . . . . 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 6223	. . . . . . 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 7848	. . . . . . . . . 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 6223	. . . . . . . 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 6213	. . . . . . . . . 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 6210	. . . . . . . . . 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 2265	. . . . . . . . 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 4118	. . . . . . . 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 801	. . . . . 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 8058	. . . . . 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 1044	. . . . 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 8058	. . . . . . 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 1043	. . . . . 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 8058	. . . . . . . 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 1042	. . . . . 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 801	. . . . 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 6857	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
78	77	rgen3 2629	. 2 ⊢ ∀𝑥 ∈ R ∀𝑦 ∈ R ∀𝑧 ∈ R (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))
79		df-iso 4417	. 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 951	1 ⊢ <R Or R

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ⟨cop 3691   class class class wbr 4108   Po wpo 4414   Or wor 4415  (class class class)co 6049  [cec 6764  Pcnp 7602   +_P cpp 7604  <_P cltp 7606   ~_R cer 7607  Rcnr 7608   <_R cltr 7614

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-enq0 7735  df-nq0 7736  df-0nq0 7737  df-plq0 7738  df-mq0 7739  df-inp 7777  df-iplp 7779  df-iltp 7781  df-enr 8037  df-nr 8038  df-ltr 8041

This theorem is referenced by:  1ne0sr  8077  addgt0sr  8086  caucvgsrlemcl  8100  caucvgsrlemfv  8102  suplocsrlemb  8117  suplocsrlempr  8118  suplocsrlem  8119  axpre-ltirr  8193  axpre-ltwlin  8194  axpre-lttrn  8195