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Theorem lttrsr 7760
Description: Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
Assertion
Ref Expression
lttrsr ((𝑓R𝑔RR) → ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R ))
Distinct variable group:   𝑓,𝑔,

Proof of Theorem lttrsr
Dummy variables 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7725 . 2 R = ((P × P) / ~R )
2 breq1 4006 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝑓 <R [⟨𝑧, 𝑤⟩] ~R ))
32anbi1d 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 4006 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R𝑓 <R [⟨𝑣, 𝑢⟩] ~R ))
53, 4imbi12d 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 4007 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 <R [⟨𝑧, 𝑤⟩] ~R𝑓 <R 𝑔))
7 breq1 4006 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R𝑔 <R [⟨𝑣, 𝑢⟩] ~R ))
86, 7anbi12d 473 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R )))
98imbi1d 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 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
10 breq2 4007 . . . 4 ([⟨𝑣, 𝑢⟩] ~R = → (𝑔 <R [⟨𝑣, 𝑢⟩] ~R𝑔 <R ))
1110anbi2d 464 . . 3 ([⟨𝑣, 𝑢⟩] ~R = → ((𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔𝑔 <R )))
12 breq2 4007 . . 3 ([⟨𝑣, 𝑢⟩] ~R = → (𝑓 <R [⟨𝑣, 𝑢⟩] ~R𝑓 <R ))
1311, 12imbi12d 234 . 2 ([⟨𝑣, 𝑢⟩] ~R = → (((𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R )))
14 ltsrprg 7745 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
15143adant3 1017 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
16 ltaprg 7617 . . . . . . . 8 ((𝑟P𝑠P𝑡P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
1716adantl 277 . . . . . . 7 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ (𝑟P𝑠P𝑡P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
18 simp1l 1021 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑥P)
19 simp2r 1024 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑤P)
20 addclpr 7535 . . . . . . . 8 ((𝑥P𝑤P) → (𝑥 +P 𝑤) ∈ P)
2118, 19, 20syl2anc 411 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 +P 𝑤) ∈ P)
22 simp1r 1022 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑦P)
23 simp2l 1023 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑧P)
24 addclpr 7535 . . . . . . . 8 ((𝑦P𝑧P) → (𝑦 +P 𝑧) ∈ P)
2522, 23, 24syl2anc 411 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 +P 𝑧) ∈ P)
26 simp3r 1026 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑢P)
27 addcomprg 7576 . . . . . . . 8 ((𝑟P𝑠P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
2827adantl 277 . . . . . . 7 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
2917, 21, 25, 26, 28caovord2d 6043 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢)))
30 addassprg 7577 . . . . . . . 8 ((𝑥P𝑤P𝑢P) → ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢)))
3118, 19, 26, 30syl3anc 1238 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢)))
32 addassprg 7577 . . . . . . . 8 ((𝑦P𝑧P𝑢P) → ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢)))
3322, 23, 26, 32syl3anc 1238 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢)))
3431, 33breq12d 4016 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
3529, 34bitrd 188 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
3615, 35bitrd 188 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
37 ltsrprg 7745 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
38373adant1 1015 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
39 addclpr 7535 . . . . . . 7 ((𝑧P𝑢P) → (𝑧 +P 𝑢) ∈ P)
4023, 26, 39syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑧 +P 𝑢) ∈ P)
41 simp3l 1025 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑣P)
42 addclpr 7535 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 +P 𝑣) ∈ P)
4319, 41, 42syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑣) ∈ P)
44 ltaprg 7617 . . . . . 6 (((𝑧 +P 𝑢) ∈ P ∧ (𝑤 +P 𝑣) ∈ P𝑦P) → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
4540, 43, 22, 44syl3anc 1238 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
4638, 45bitrd 188 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
4736, 46anbi12d 473 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (([⟨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 7594 . . . . 5 <P Or P
49 ltrelpr 7503 . . . . 5 <P ⊆ (P × P)
5048, 49sotri 5024 . . . 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 7535 . . . . . . . 8 ((𝑥P𝑢P) → (𝑥 +P 𝑢) ∈ P)
5218, 26, 51syl2anc 411 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 +P 𝑢) ∈ P)
53 addclpr 7535 . . . . . . . 8 ((𝑦P𝑣P) → (𝑦 +P 𝑣) ∈ P)
5422, 41, 53syl2anc 411 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 +P 𝑣) ∈ P)
55 ltaprg 7617 . . . . . . 7 (((𝑥 +P 𝑢) ∈ P ∧ (𝑦 +P 𝑣) ∈ P𝑤P) → ((𝑥 +P 𝑢)<P (𝑦 +P 𝑣) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
5652, 54, 19, 55syl3anc 1238 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑥 +P 𝑢)<P (𝑦 +P 𝑣) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
5756biimprd 158 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)) → (𝑥 +P 𝑢)<P (𝑦 +P 𝑣)))
58 addassprg 7577 . . . . . . . 8 ((𝑟P𝑠P𝑡P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
5958adantl 277 . . . . . . 7 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ (𝑟P𝑠P𝑡P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
6018, 19, 26, 28, 59caov12d 6055 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 +P (𝑤 +P 𝑢)) = (𝑤 +P (𝑥 +P 𝑢)))
6122, 19, 41, 28, 59caov12d 6055 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 +P (𝑤 +P 𝑣)) = (𝑤 +P (𝑦 +P 𝑣)))
6260, 61breq12d 4016 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
63 ltsrprg 7745 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣)))
64633adant2 1016 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣)))
6557, 62, 643imtr4d 203 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
6650, 65syl5 32 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (((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 ))
6747, 66sylbid 150 . 2 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
681, 5, 9, 13, 673ecoptocl 6623 1 ((𝑓R𝑔RR) → ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  cop 3595   class class class wbr 4003  (class class class)co 5874  [cec 6532  Pcnp 7289   +P cpp 7291  <P cltp 7293   ~R cer 7294  Rcnr 7295   <R cltr 7301
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468  df-enr 7724  df-nr 7725  df-ltr 7728
This theorem is referenced by:  ltposr  7761
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