Theorem lttrsr 7724

Description: Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)

Assertion

Ref	Expression
lttrsr	⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ))

Distinct variable group:   𝑓,𝑔,ℎ

Proof of Theorem lttrsr

Dummy variables 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7689	. 2 ⊢ R = ((P × P) / ~R )
2		breq1 3992	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R [⟨𝑧, 𝑤⟩] ~R ))
3	2	anbi1d 462	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )))
4		breq1 3992	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ))
5	3, 4	imbi12d 233	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ((([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
6		breq2 3993	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R 𝑔))
7		breq1 3992	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ))
8	6, 7	anbi12d 470	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R )))
9	8	imbi1d 230	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
10		breq2 3993	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑔 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R ℎ))
11	10	anbi2d 461	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → ((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔 ∧ 𝑔 <R ℎ)))
12		breq2 3993	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑓 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R ℎ))
13	11, 12	imbi12d 233	. 2 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ)))
14		ltsrprg 7709	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
15	14	3adant3 1012	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
16		ltaprg 7581	. . . . . . . 8 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
17	16	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ (𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
18		simp1l 1016	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑥 ∈ P)
19		simp2r 1019	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑤 ∈ P)
20		addclpr 7499	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) ∈ P)
21	18, 19, 20	syl2anc 409	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 +P 𝑤) ∈ P)
22		simp1r 1017	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑦 ∈ P)
23		simp2l 1018	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑧 ∈ P)
24		addclpr 7499	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) ∈ P)
25	22, 23, 24	syl2anc 409	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 +P 𝑧) ∈ P)
26		simp3r 1021	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑢 ∈ P)
27		addcomprg 7540	. . . . . . . 8 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
28	27	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ (𝑟 ∈ P ∧ 𝑠 ∈ P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
29	17, 21, 25, 26, 28	caovord2d 6022	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢)))
30		addassprg 7541	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢)))
31	18, 19, 26, 30	syl3anc 1233	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢)))
32		addassprg 7541	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P ∧ 𝑢 ∈ P) → ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢)))
33	22, 23, 26, 32	syl3anc 1233	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢)))
34	31, 33	breq12d 4002	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
35	29, 34	bitrd 187	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
36	15, 35	bitrd 187	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
37		ltsrprg 7709	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
38	37	3adant1 1010	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
39		addclpr 7499	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑢 ∈ P) → (𝑧 +P 𝑢) ∈ P)
40	23, 26, 39	syl2anc 409	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑧 +P 𝑢) ∈ P)
41		simp3l 1020	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑣 ∈ P)
42		addclpr 7499	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 +P 𝑣) ∈ P)
43	19, 41, 42	syl2anc 409	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑣) ∈ P)
44		ltaprg 7581	. . . . . 6 ⊢ (((𝑧 +P 𝑢) ∈ P ∧ (𝑤 +P 𝑣) ∈ P ∧ 𝑦 ∈ P) → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
45	40, 43, 22, 44	syl3anc 1233	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
46	38, 45	bitrd 187	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
47	36, 46	anbi12d 470	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)))))
48		ltsopr 7558	. . . . 5 ⊢ <P Or P
49		ltrelpr 7467	. . . . 5 ⊢ <P ⊆ (P × P)
50	48, 49	sotri 5006	. . . 4 ⊢ (((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))) → (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)))
51		addclpr 7499	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑢 ∈ P) → (𝑥 +P 𝑢) ∈ P)
52	18, 26, 51	syl2anc 409	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 +P 𝑢) ∈ P)
53		addclpr 7499	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 +P 𝑣) ∈ P)
54	22, 41, 53	syl2anc 409	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 +P 𝑣) ∈ P)
55		ltaprg 7581	. . . . . . 7 ⊢ (((𝑥 +P 𝑢) ∈ P ∧ (𝑦 +P 𝑣) ∈ P ∧ 𝑤 ∈ P) → ((𝑥 +P 𝑢)<P (𝑦 +P 𝑣) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
56	52, 54, 19, 55	syl3anc 1233	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P 𝑢)<P (𝑦 +P 𝑣) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
57	56	biimprd 157	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)) → (𝑥 +P 𝑢)<P (𝑦 +P 𝑣)))
58		addassprg 7541	. . . . . . . 8 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
59	58	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ (𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
60	18, 19, 26, 28, 59	caov12d 6034	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 +P (𝑤 +P 𝑢)) = (𝑤 +P (𝑥 +P 𝑢)))
61	22, 19, 41, 28, 59	caov12d 6034	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 +P (𝑤 +P 𝑣)) = (𝑤 +P (𝑦 +P 𝑣)))
62	60, 61	breq12d 4002	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
63		ltsrprg 7709	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣)))
64	63	3adant2 1011	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣)))
65	57, 62, 64	3imtr4d 202	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
66	50, 65	syl5 32	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
67	47, 66	sylbid 149	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
68	1, 5, 9, 13, 67	3ecoptocl 6602	1 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ⟨cop 3586   class class class wbr 3989  (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:  ltposr  7725