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Theorem noinfbnd1 27077
Description: Bounding law from above for the surreal infimum. Analagous to proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
noinfbnd1.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfbnd1 ((𝐵 No 𝐵𝑉𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑥,𝑢,𝑦   𝑔,𝑉   𝑥,𝑣,𝑦,𝑈   𝑥,𝑉
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑢,𝑔)   𝑉(𝑦,𝑣,𝑢)

Proof of Theorem noinfbnd1
StepHypRef Expression
1 simpr1 1194 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
2 simpl 483 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3 nominmo 27047 . . . . . . . . 9 (𝐵 No → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
41, 3syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
5 reu5 3355 . . . . . . . 8 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
62, 4, 5sylanbrc 583 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
7 riotacl 7331 . . . . . . 7 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
86, 7syl 17 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
91, 8sseldd 3945 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
10 noextendlt 27017 . . . . 5 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
119, 10syl 17 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
12 simpr3 1196 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈𝐵)
13 nfv 1917 . . . . . . . . 9 𝑥(𝐵 No 𝐵𝑉𝑈𝐵)
14 nfcv 2907 . . . . . . . . . 10 𝑥𝐵
15 nfcv 2907 . . . . . . . . . . . 12 𝑥𝑦
16 nfcv 2907 . . . . . . . . . . . 12 𝑥 <s
17 nfriota1 7320 . . . . . . . . . . . 12 𝑥(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1815, 16, 17nfbr 5152 . . . . . . . . . . 11 𝑥 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1918nfn 1860 . . . . . . . . . 10 𝑥 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2014, 19nfralw 3294 . . . . . . . . 9 𝑥𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2113, 20nfim 1899 . . . . . . . 8 𝑥((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
22 simpl 483 . . . . . . . . . . 11 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
23 rspe 3232 . . . . . . . . . . . . . 14 ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2423adantr 481 . . . . . . . . . . . . 13 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
25 simpr1 1194 . . . . . . . . . . . . . 14 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
2625, 3syl 17 . . . . . . . . . . . . 13 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2724, 26, 5sylanbrc 583 . . . . . . . . . . . 12 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
28 riota1 7335 . . . . . . . . . . . 12 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥))
2927, 28syl 17 . . . . . . . . . . 11 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥))
3022, 29mpbid 231 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥)
31 simplr 767 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s 𝑥)
32 nfra1 3267 . . . . . . . . . . . . . 14 𝑦𝑦𝐵 ¬ 𝑦 <s 𝑥
33 nfcv 2907 . . . . . . . . . . . . . 14 𝑦𝐵
3432, 33nfriota 7326 . . . . . . . . . . . . 13 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3534nfeq1 2922 . . . . . . . . . . . 12 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥
36 breq2 5109 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑦 <s 𝑥))
3736notbid 317 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑦 <s 𝑥))
3835, 37ralbid 3256 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
3938biimprd 247 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4030, 31, 39sylc 65 . . . . . . . . 9 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
4140exp31 420 . . . . . . . 8 (𝑥𝐵 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
4221, 41rexlimi 3242 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4342imp 407 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
44 nfcv 2907 . . . . . . . . 9 𝑦𝑈
45 nfcv 2907 . . . . . . . . 9 𝑦 <s
4644, 45, 34nfbr 5152 . . . . . . . 8 𝑦 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
4746nfn 1860 . . . . . . 7 𝑦 ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
48 breq1 5108 . . . . . . . 8 (𝑦 = 𝑈 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4948notbid 317 . . . . . . 7 (𝑦 = 𝑈 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
5047, 49rspc 3569 . . . . . 6 (𝑈𝐵 → (∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) → ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
5112, 43, 50sylc 65 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
52 nofun 26997 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → Fun (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
53 funrel 6518 . . . . . . . . 9 (Fun (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) → Rel (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
549, 52, 533syl 18 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → Rel (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
55 sssucid 6397 . . . . . . . 8 dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
56 relssres 5978 . . . . . . . 8 ((Rel (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
5754, 55, 56sylancl 586 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
5857breq2d 5117 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ↔ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
591, 12sseldd 3945 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈 No )
60 nodmon 26998 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
619, 60syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
62 onsucb 7752 . . . . . . . 8 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On ↔ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
6361, 62sylib 217 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
64 sltres 27010 . . . . . . 7 ((𝑈 No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No ∧ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
6559, 9, 63, 64syl3anc 1371 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
6658, 65sylbird 259 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) → 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
6751, 66mtod 197 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
68 1oex 8422 . . . . . . . 8 1o ∈ V
6968prid1 4723 . . . . . . 7 1o ∈ {1o, 2o}
7069noextend 27014 . . . . . 6 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No )
719, 70syl 17 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No )
72 noreson 27008 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
7359, 63, 72syl2anc 584 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
74 sltso 27024 . . . . . 6 <s Or No
75 sotr3 5584 . . . . . 6 (( <s Or No ∧ (((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No ∧ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )) → ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
7674, 75mpan 688 . . . . 5 ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No ∧ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No ) → ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
7771, 9, 73, 76syl3anc 1371 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
7811, 67, 77mp2and 697 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
79 noinfbnd1.1 . . . . 5 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
80 iftrue 4492 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8179, 80eqtrid 2788 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8281adantr 481 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8381dmeqd 5861 . . . . . 6 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8468dmsnop 6168 . . . . . . . 8 dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)}
8584uneq2i 4120 . . . . . . 7 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
86 dmun 5866 . . . . . . 7 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩})
87 df-suc 6323 . . . . . . 7 suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
8885, 86, 873eqtr4i 2774 . . . . . 6 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
8983, 88eqtrdi 2792 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
9089reseq2d 5937 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9190adantr 481 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9278, 82, 913brtr4d 5137 . 2 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
93 simpl 483 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
94 simpr1 1194 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
95 simpr2 1195 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵𝑉)
96 simpr3 1196 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈𝐵)
9779noinfbnd1lem6 27076 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ 𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
9893, 94, 95, 96, 97syl121anc 1375 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
9992, 98pm2.61ian 810 1 ((𝐵 No 𝐵𝑉𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  ∃!wreu 3351  ∃*wrmo 3352  cun 3908  wss 3910  ifcif 4486  {csn 4586  cop 4592   class class class wbr 5105  cmpt 5188   Or wor 5544  dom cdm 5633  cres 5635  Rel wrel 5638  Oncon0 6317  suc csuc 6319  cio 6446  Fun wfun 6490  cfv 6496  crio 7312  1oc1o 8405  2oc2o 8406   No csur 26988   <s cslt 26989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993
This theorem is referenced by:  noinfbnd2  27079  noetainflem3  27087
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