Theorem lttrsr 7822

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 7787	. 2 ⊢ R = ((P × P) / ~R )
2		breq1 4032	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R [⟨𝑧, 𝑤⟩] ~R ))
3	2	anbi1d 465	. . 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 4032	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ))
5	3, 4	imbi12d 234	. 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 4033	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑓 <R 𝑔))
7		breq1 4032	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ))
8	6, 7	anbi12d 473	. . 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 231	. 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 4033	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑔 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑔 <R ℎ))
11	10	anbi2d 464	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → ((𝑓 <R 𝑔 ∧ 𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔 ∧ 𝑔 <R ℎ)))
12		breq2 4033	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ~R = ℎ → (𝑓 <R [⟨𝑣, 𝑢⟩] ~R ↔ 𝑓 <R ℎ))
13	11, 12	imbi12d 234	. 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 7807	. . . . . 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 1019	. . . . 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 7679	. . . . . . . 8 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
17	16	adantl 277	. . . . . . 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 1023	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑥 ∈ P)
19		simp2r 1026	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑤 ∈ P)
20		addclpr 7597	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) ∈ P)
21	18, 19, 20	syl2anc 411	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 +P 𝑤) ∈ P)
22		simp1r 1024	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑦 ∈ P)
23		simp2l 1025	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑧 ∈ P)
24		addclpr 7597	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) ∈ P)
25	22, 23, 24	syl2anc 411	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 +P 𝑧) ∈ P)
26		simp3r 1028	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑢 ∈ P)
27		addcomprg 7638	. . . . . . . 8 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
28	27	adantl 277	. . . . . . 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 6088	. . . . . 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 7639	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢)))
31	18, 19, 26, 30	syl3anc 1249	. . . . . . 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 7639	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P ∧ 𝑢 ∈ P) → ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢)))
33	22, 23, 26, 32	syl3anc 1249	. . . . . . 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 4042	. . . . . 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 188	. . . . 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 188	. . . 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 7807	. . . . . 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 1017	. . . . 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 7597	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑢 ∈ P) → (𝑧 +P 𝑢) ∈ P)
40	23, 26, 39	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑧 +P 𝑢) ∈ P)
41		simp3l 1027	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑣 ∈ P)
42		addclpr 7597	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 +P 𝑣) ∈ P)
43	19, 41, 42	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑣) ∈ P)
44		ltaprg 7679	. . . . . 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 1249	. . . . 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 188	. . . 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 473	. . 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 7656	. . . . 5 ⊢ <P Or P
49		ltrelpr 7565	. . . . 5 ⊢ <P ⊆ (P × P)
50	48, 49	sotri 5061	. . . 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 7597	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑢 ∈ P) → (𝑥 +P 𝑢) ∈ P)
52	18, 26, 51	syl2anc 411	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 +P 𝑢) ∈ P)
53		addclpr 7597	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 +P 𝑣) ∈ P)
54	22, 41, 53	syl2anc 411	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 +P 𝑣) ∈ P)
55		ltaprg 7679	. . . . . . 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 1249	. . . . . 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 158	. . . . 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 7639	. . . . . . . 8 ⊢ ((𝑟 ∈ P ∧ 𝑠 ∈ P ∧ 𝑡 ∈ P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
59	58	adantl 277	. . . . . . 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 6100	. . . . . 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 6100	. . . . . 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 4042	. . . . 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 7807	. . . . . 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 1018	. . . . 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 203	. . . 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 150	. 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 6678	1 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  (class class class)co 5918  [cec 6585  Pcnp 7351   +_P cpp 7353  <_P cltp 7355   ~_R cer 7356  Rcnr 7357   <_R cltr 7363

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530  df-enr 7786  df-nr 7787  df-ltr 7790

This theorem is referenced by:  ltposr  7823