Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift2lem12 Structured version   Visualization version   GIF version

Theorem cvmlift2lem12 31039
Description: Lemma for cvmlift2 31041. (Contributed by Mario Carneiro, 1-Jun-2015.)
Hypotheses
Ref Expression
cvmlift2.b 𝐵 = 𝐶
cvmlift2.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift2.g (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))
cvmlift2.p (𝜑𝑃𝐵)
cvmlift2.i (𝜑 → (𝐹𝑃) = (0𝐺0))
cvmlift2.h 𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
cvmlift2.k 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))
cvmlift2.m 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}
cvmlift2.a 𝐴 = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀}
cvmlift2.s 𝑆 = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))}
Assertion
Ref Expression
cvmlift2lem12 (𝜑𝐾 ∈ ((II ×t II) Cn 𝐶))
Distinct variable groups:   𝑢,𝑓,𝑥,𝑦,𝑧,𝐹   𝑓,𝑎,𝑟,𝑡,𝑢,𝑥,𝑦,𝑧,𝜑   𝐴,𝑎,𝑡,𝑥   𝑀,𝑎,𝑟,𝑢,𝑥,𝑦,𝑧   𝑆,𝑓,𝑡,𝑢,𝑥,𝑦,𝑧   𝑓,𝐽,𝑢,𝑥,𝑦,𝑧   𝐺,𝑎,𝑓,𝑡,𝑢,𝑥,𝑦,𝑧   𝑓,𝐻,𝑢,𝑥,𝑦,𝑧   𝐶,𝑎,𝑓,𝑟,𝑡,𝑢,𝑥,𝑦,𝑧   𝑃,𝑓,𝑢,𝑥,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝐾,𝑎,𝑓,𝑟,𝑡,𝑢,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑢,𝑓,𝑟)   𝐵(𝑢,𝑡,𝑓,𝑟,𝑎)   𝑃(𝑡,𝑟,𝑎)   𝑆(𝑟,𝑎)   𝐹(𝑡,𝑟,𝑎)   𝐺(𝑟)   𝐻(𝑡,𝑟,𝑎)   𝐽(𝑡,𝑟,𝑎)   𝑀(𝑡,𝑓)

Proof of Theorem cvmlift2lem12
Dummy variables 𝑏 𝑐 𝑑 𝑘 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift2.b . . 3 𝐵 = 𝐶
2 cvmlift2.f . . 3 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
3 cvmlift2.g . . 3 (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))
4 cvmlift2.p . . 3 (𝜑𝑃𝐵)
5 cvmlift2.i . . 3 (𝜑 → (𝐹𝑃) = (0𝐺0))
6 cvmlift2.h . . 3 𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
7 cvmlift2.k . . 3 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))
81, 2, 3, 4, 5, 6, 7cvmlift2lem5 31032 . 2 (𝜑𝐾:((0[,]1) × (0[,]1))⟶𝐵)
9 iunid 4546 . . . . . . 7 𝑎 ∈ (0[,]1){𝑎} = (0[,]1)
109xpeq2i 5101 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = ((0[,]1) × (0[,]1))
11 xpiundi 5139 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
1210, 11eqtr3i 2645 . . . . 5 ((0[,]1) × (0[,]1)) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
13 cvmlift2.a . . . . . . . 8 𝐴 = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀}
14 iiuni 22607 . . . . . . . . 9 (0[,]1) = II
15 iiconn 22613 . . . . . . . . . 10 II ∈ Conn
1615a1i 11 . . . . . . . . 9 (𝜑 → II ∈ Conn)
17 inss1 3816 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ II
18 iicmp 22612 . . . . . . . . . . . . . . 15 II ∈ Comp
1918a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Comp)
20 iitop 22606 . . . . . . . . . . . . . . 15 II ∈ Top
2120a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Top)
2220, 20txtopi 21316 . . . . . . . . . . . . . . . 16 (II ×t II) ∈ Top
2314neiss2 20828 . . . . . . . . . . . . . . . . . . . . . . . 24 ((II ∈ Top ∧ 𝑢 ∈ ((nei‘II)‘{𝑟})) → {𝑟} ⊆ (0[,]1))
2420, 23mpan 705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ ((nei‘II)‘{𝑟}) → {𝑟} ⊆ (0[,]1))
25 vex 3192 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑟 ∈ V
2625snss 4291 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ (0[,]1) ↔ {𝑟} ⊆ (0[,]1))
2724, 26sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 ∈ ((nei‘II)‘{𝑟}) → 𝑟 ∈ (0[,]1))
2827a1d 25 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ ((nei‘II)‘{𝑟}) → (((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → 𝑟 ∈ (0[,]1)))
2928rexlimiv 3021 . . . . . . . . . . . . . . . . . . . 20 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → 𝑟 ∈ (0[,]1))
3029adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → 𝑟 ∈ (0[,]1))
31 simpl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → 𝑡 ∈ (0[,]1))
3230, 31jca 554 . . . . . . . . . . . . . . . . . 18 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1)))
3332ssopab2i 4968 . . . . . . . . . . . . . . . . 17 {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))}
34 cvmlift2.s . . . . . . . . . . . . . . . . 17 𝑆 = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))}
35 df-xp 5085 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))}
3633, 34, 353sstr4i 3628 . . . . . . . . . . . . . . . 16 𝑆 ⊆ ((0[,]1) × (0[,]1))
3720, 20, 14, 14txunii 21319 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = (II ×t II)
3837ntropn 20776 . . . . . . . . . . . . . . . 16 (((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) → ((int‘(II ×t II))‘𝑆) ∈ (II ×t II))
3922, 36, 38mp2an 707 . . . . . . . . . . . . . . 15 ((int‘(II ×t II))‘𝑆) ∈ (II ×t II)
4039a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → ((int‘(II ×t II))‘𝑆) ∈ (II ×t II))
412adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
423adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
434adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑃𝐵)
445adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → (𝐹𝑃) = (0𝐺0))
45 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))}) = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
46 simprr 795 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑏 ∈ (0[,]1))
47 simprl 793 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑎 ∈ (0[,]1))
481, 41, 42, 43, 44, 6, 7, 45, 46, 47cvmlift2lem10 31037 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
4922a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → (II ×t II) ∈ Top)
5036a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → 𝑆 ⊆ ((0[,]1) × (0[,]1)))
5120a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → II ∈ Top)
52 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → 𝑢 ∈ II)
53 simplrr 800 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → 𝑣 ∈ II)
54 txopn 21328 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ∈ Top ∧ II ∈ Top) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → (𝑢 × 𝑣) ∈ (II ×t II))
5551, 51, 52, 53, 54syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → (𝑢 × 𝑣) ∈ (II ×t II))
56 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑟𝑢𝑡𝑣) → 𝑡𝑣)
57 elunii 4412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑡𝑣𝑣 ∈ II) → 𝑡 II)
5857, 14syl6eleqr 2709 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑡𝑣𝑣 ∈ II) → 𝑡 ∈ (0[,]1))
5956, 53, 58syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑡 ∈ (0[,]1))
6020a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → II ∈ Top)
6152adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑢 ∈ II)
62 simprl 793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑟𝑢)
63 opnneip 20846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((II ∈ Top ∧ 𝑢 ∈ II ∧ 𝑟𝑢) → 𝑢 ∈ ((nei‘II)‘{𝑟}))
6460, 61, 62, 63syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑢 ∈ ((nei‘II)‘{𝑟}))
6541ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
6642ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
6743ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑃𝐵)
6844ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → (𝐹𝑃) = (0𝐺0))
69 cvmlift2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}
7053adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑣 ∈ II)
71 simplr2 1102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑎𝑣)
72 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑡𝑣)
73 sneq 4163 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → {𝑐} = {𝑤})
7473xpeq2d 5104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (𝑢 × {𝑐}) = (𝑢 × {𝑤}))
7574reseq2d 5361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (𝐾 ↾ (𝑢 × {𝑐})) = (𝐾 ↾ (𝑢 × {𝑤})))
7674oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → ((II ×t II) ↾t (𝑢 × {𝑐})) = ((II ×t II) ↾t (𝑢 × {𝑤})))
7776oveq1d 6625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
7875, 77eleq12d 2692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑐 = 𝑤 → ((𝐾 ↾ (𝑢 × {𝑐})) ∈ (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) ↔ (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶)))
7978cbvrexv 3163 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑐𝑣 (𝐾 ↾ (𝑢 × {𝑐})) ∈ (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) ↔ ∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
80 simplr3 1103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))
8179, 80syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → (∃𝑐𝑣 (𝐾 ↾ (𝑢 × {𝑐})) ∈ (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))
821, 65, 66, 67, 68, 6, 7, 69, 61, 70, 71, 72, 81cvmlift2lem11 31038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → ((𝑢 × {𝑎}) ⊆ 𝑀 → (𝑢 × {𝑡}) ⊆ 𝑀))
831, 65, 66, 67, 68, 6, 7, 69, 61, 70, 72, 71, 81cvmlift2lem11 31038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → ((𝑢 × {𝑡}) ⊆ 𝑀 → (𝑢 × {𝑎}) ⊆ 𝑀))
8482, 83impbid 202 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → ((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
85 rspe 2998 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑢 ∈ ((nei‘II)‘{𝑟}) ∧ ((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
8664, 84, 85syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
8759, 86jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
8887ex 450 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → ((𝑟𝑢𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
8988alrimivv 1853 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → ∀𝑟𝑡((𝑟𝑢𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
90 df-xp 5085 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)}
9190, 34sseq12i 3615 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))})
92 ssopab2b 4967 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))} ↔ ∀𝑟𝑡((𝑟𝑢𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
9391, 92bitri 264 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ ∀𝑟𝑡((𝑟𝑢𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
9489, 93sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → (𝑢 × 𝑣) ⊆ 𝑆)
9537ssntr 20785 . . . . . . . . . . . . . . . . . . . . . . 23 ((((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) ∧ ((𝑢 × 𝑣) ∈ (II ×t II) ∧ (𝑢 × 𝑣) ⊆ 𝑆)) → (𝑢 × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆))
9649, 50, 55, 94, 95syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → (𝑢 × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆))
97 simpr1 1065 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → 𝑏𝑢)
98 simpr2 1066 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → 𝑎𝑣)
99 opelxpi 5113 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑢𝑎𝑣) → ⟨𝑏, 𝑎⟩ ∈ (𝑢 × 𝑣))
10097, 98, 99syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → ⟨𝑏, 𝑎⟩ ∈ (𝑢 × 𝑣))
10196, 100sseldd 3588 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) → ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆))
102101ex 450 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → ((𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))) → ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆)))
103102rexlimdvva 3032 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → (∃𝑢 ∈ II ∃𝑣 ∈ II (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))) → ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆)))
10448, 103mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆))
105 vex 3192 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
106 opeq2 4376 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑎 → ⟨𝑏, 𝑤⟩ = ⟨𝑏, 𝑎⟩)
107106eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑎 → (⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆)))
108105, 107ralsn 4198 . . . . . . . . . . . . . . . . . 18 (∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆))
109104, 108sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → ∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
110109anassrs 679 . . . . . . . . . . . . . . . 16 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑏 ∈ (0[,]1)) → ∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
111110ralrimiva 2961 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (0[,]1)) → ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
112 dfss3 3577 . . . . . . . . . . . . . . . 16 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑢 ∈ ((0[,]1) × {𝑎})𝑢 ∈ ((int‘(II ×t II))‘𝑆))
113 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑏, 𝑤⟩ → (𝑢 ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆)))
114113ralxp 5228 . . . . . . . . . . . . . . . 16 (∀𝑢 ∈ ((0[,]1) × {𝑎})𝑢 ∈ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
115112, 114bitri 264 . . . . . . . . . . . . . . 15 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
116111, 115sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → ((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆))
117 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → 𝑎 ∈ (0[,]1))
11814, 14, 19, 21, 40, 116, 117txtube 21366 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (0[,]1)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)))
11937ntrss2 20784 . . . . . . . . . . . . . . . . . . 19 (((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) → ((int‘(II ×t II))‘𝑆) ⊆ 𝑆)
12022, 36, 119mp2an 707 . . . . . . . . . . . . . . . . . 18 ((int‘(II ×t II))‘𝑆) ⊆ 𝑆
121 sstr 3595 . . . . . . . . . . . . . . . . . 18 ((((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) ∧ ((int‘(II ×t II))‘𝑆) ⊆ 𝑆) → ((0[,]1) × 𝑣) ⊆ 𝑆)
122120, 121mpan2 706 . . . . . . . . . . . . . . . . 17 (((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) → ((0[,]1) × 𝑣) ⊆ 𝑆)
123 df-xp 5085 . . . . . . . . . . . . . . . . . . 19 ((0[,]1) × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)}
124123, 34sseq12i 3615 . . . . . . . . . . . . . . . . . 18 (((0[,]1) × 𝑣) ⊆ 𝑆 ↔ {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))})
125 ssopab2b 4967 . . . . . . . . . . . . . . . . . . 19 ({⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))} ↔ ∀𝑟𝑡((𝑟 ∈ (0[,]1) ∧ 𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
126 r2al 2934 . . . . . . . . . . . . . . . . . . 19 (∀𝑟 ∈ (0[,]1)∀𝑡𝑣 (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) ↔ ∀𝑟𝑡((𝑟 ∈ (0[,]1) ∧ 𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
127 ralcom 3091 . . . . . . . . . . . . . . . . . . 19 (∀𝑟 ∈ (0[,]1)∀𝑡𝑣 (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) ↔ ∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
128125, 126, 1273bitr2i 288 . . . . . . . . . . . . . . . . . 18 ({⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))} ↔ ∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
129124, 128bitri 264 . . . . . . . . . . . . . . . . 17 (((0[,]1) × 𝑣) ⊆ 𝑆 ↔ ∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
130122, 129sylib 208 . . . . . . . . . . . . . . . 16 (((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) → ∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
131 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
132131ralimi 2947 . . . . . . . . . . . . . . . . . . 19 (∀𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
133 cvmlift2lem1 31027 . . . . . . . . . . . . . . . . . . . 20 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑎}) ⊆ 𝑀 → ((0[,]1) × {𝑡}) ⊆ 𝑀))
134 bicom 212 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
135134rexbii 3035 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
136135ralbii 2975 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
137 cvmlift2lem1 31027 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀) → (((0[,]1) × {𝑡}) ⊆ 𝑀 → ((0[,]1) × {𝑎}) ⊆ 𝑀))
138136, 137sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑡}) ⊆ 𝑀 → ((0[,]1) × {𝑎}) ⊆ 𝑀))
139133, 138impbid 202 . . . . . . . . . . . . . . . . . . 19 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
140132, 139syl 17 . . . . . . . . . . . . . . . . . 18 (∀𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
14113rabeq2i 3186 . . . . . . . . . . . . . . . . . . . . 21 (𝑎𝐴 ↔ (𝑎 ∈ (0[,]1) ∧ ((0[,]1) × {𝑎}) ⊆ 𝑀))
142141baib 943 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (0[,]1) → (𝑎𝐴 ↔ ((0[,]1) × {𝑎}) ⊆ 𝑀))
143142ad3antlr 766 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → (𝑎𝐴 ↔ ((0[,]1) × {𝑎}) ⊆ 𝑀))
144 elssuni 4438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ II → 𝑣 II)
145144, 14syl6sseqr 3636 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ II → 𝑣 ⊆ (0[,]1))
146145adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → 𝑣 ⊆ (0[,]1))
147146sselda 3587 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → 𝑡 ∈ (0[,]1))
148 sneq 4163 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑡 → {𝑎} = {𝑡})
149148xpeq2d 5104 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑡 → ((0[,]1) × {𝑎}) = ((0[,]1) × {𝑡}))
150149sseq1d 3616 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑡 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
151150, 13elrab2 3352 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐴 ↔ (𝑡 ∈ (0[,]1) ∧ ((0[,]1) × {𝑡}) ⊆ 𝑀))
152151baib 943 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ (0[,]1) → (𝑡𝐴 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
153147, 152syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → (𝑡𝐴 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
154143, 153bibi12d 335 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → ((𝑎𝐴𝑡𝐴) ↔ (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀)))
155140, 154syl5ibr 236 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → (∀𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → (𝑎𝐴𝑡𝐴)))
156155ralimdva 2957 . . . . . . . . . . . . . . . 16 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → (∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
157130, 156syl5 34 . . . . . . . . . . . . . . 15 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → (((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) → ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
158157anim2d 588 . . . . . . . . . . . . . 14 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → ((𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)) → (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
159158reximdva 3012 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (0[,]1)) → (∃𝑣 ∈ II (𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
160118, 159mpd 15 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (0[,]1)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
161160ralrimiva 2961 . . . . . . . . . . 11 (𝜑 → ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
162 ssrab2 3671 . . . . . . . . . . . . 13 {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ⊆ (0[,]1)
16313, 162eqsstri 3619 . . . . . . . . . . . 12 𝐴 ⊆ (0[,]1)
16414isclo 20814 . . . . . . . . . . . 12 ((II ∈ Top ∧ 𝐴 ⊆ (0[,]1)) → (𝐴 ∈ (II ∩ (Clsd‘II)) ↔ ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
16520, 163, 164mp2an 707 . . . . . . . . . . 11 (𝐴 ∈ (II ∩ (Clsd‘II)) ↔ ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
166161, 165sylibr 224 . . . . . . . . . 10 (𝜑𝐴 ∈ (II ∩ (Clsd‘II)))
16717, 166sseldi 3585 . . . . . . . . 9 (𝜑𝐴 ∈ II)
168 0elunit 12240 . . . . . . . . . . . 12 0 ∈ (0[,]1)
169168a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ∈ (0[,]1))
170 relxp 5193 . . . . . . . . . . . . 13 Rel ((0[,]1) × {0})
171170a1i 11 . . . . . . . . . . . 12 (𝜑 → Rel ((0[,]1) × {0}))
172 opelxp 5111 . . . . . . . . . . . . 13 (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) ↔ (𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}))
173 id 22 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ (0[,]1) → 𝑟 ∈ (0[,]1))
174 opelxpi 5113 . . . . . . . . . . . . . . . . 17 ((𝑟 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑟, 0⟩ ∈ ((0[,]1) × (0[,]1)))
175173, 169, 174syl2anr 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ (0[,]1)) → ⟨𝑟, 0⟩ ∈ ((0[,]1) × (0[,]1)))
1762adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
1773adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
1784adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝑃𝐵)
1795adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → (𝐹𝑃) = (0𝐺0))
180 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝑟 ∈ (0[,]1))
181168a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 0 ∈ (0[,]1))
1821, 176, 177, 178, 179, 6, 7, 45, 180, 181cvmlift2lem10 31037 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟 ∈ (0[,]1)) → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑟𝑢 ∧ 0 ∈ 𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
183 df-3an 1038 . . . . . . . . . . . . . . . . . . 19 ((𝑟𝑢 ∧ 0 ∈ 𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))) ↔ ((𝑟𝑢 ∧ 0 ∈ 𝑣) ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
184 simprr 795 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 0 ∈ 𝑣)
1858ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
186 ffn 6007 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐾:((0[,]1) × (0[,]1))⟶𝐵𝐾 Fn ((0[,]1) × (0[,]1)))
187185, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾 Fn ((0[,]1) × (0[,]1)))
188 fnov 6728 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐾 Fn ((0[,]1) × (0[,]1)) ↔ 𝐾 = (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)))
189187, 188sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾 = (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)))
190189reseq1d 5360 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) = ((𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)) ↾ (𝑢 × {0})))
191 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ∈ II)
192 elssuni 4438 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 ∈ II → 𝑢 II)
193192, 14syl6sseqr 3636 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 ∈ II → 𝑢 ⊆ (0[,]1))
194191, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ⊆ (0[,]1))
195169snssd 4314 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → {0} ⊆ (0[,]1))
196195ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → {0} ⊆ (0[,]1))
197 resmpt2 6718 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 ⊆ (0[,]1) ∧ {0} ⊆ (0[,]1)) → ((𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)) ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)))
198194, 196, 197syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)) ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)))
199194sselda 3587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → 𝑏 ∈ (0[,]1))
200 simplll 797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝜑)
2011, 2, 3, 4, 5, 6, 7cvmlift2lem8 31035 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑏 ∈ (0[,]1)) → (𝑏𝐾0) = (𝐻𝑏))
202200, 201sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏 ∈ (0[,]1)) → (𝑏𝐾0) = (𝐻𝑏))
203199, 202syldan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → (𝑏𝐾0) = (𝐻𝑏))
204 elsni 4170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 ∈ {0} → 𝑤 = 0)
205204oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {0} → (𝑏𝐾𝑤) = (𝑏𝐾0))
206205eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ {0} → ((𝑏𝐾𝑤) = (𝐻𝑏) ↔ (𝑏𝐾0) = (𝐻𝑏)))
207203, 206syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → (𝑤 ∈ {0} → (𝑏𝐾𝑤) = (𝐻𝑏)))
2082073impia 1258 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢𝑤 ∈ {0}) → (𝑏𝐾𝑤) = (𝐻𝑏))
209208mpt2eq3dva 6679 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
210190, 198, 2093eqtrd 2659 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
211 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t 𝑢) = (II ↾t 𝑢)
212 iitopon 22605 . . . . . . . . . . . . . . . . . . . . . . . . . 26 II ∈ (TopOn‘(0[,]1))
213212a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → II ∈ (TopOn‘(0[,]1)))
214 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t {0}) = (II ↾t {0})
215213, 213cnmpt1st 21394 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ 𝑏) ∈ ((II ×t II) Cn II))
2161, 2, 3, 4, 5, 6cvmlift2lem2 31029 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
217216simp1d 1071 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐻 ∈ (II Cn 𝐶))
218200, 217syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐻 ∈ (II Cn 𝐶))
219213, 213, 215, 218cnmpt21f 21398 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝐻𝑏)) ∈ ((II ×t II) Cn 𝐶))
220211, 213, 194, 214, 213, 196, 219cnmpt2res 21403 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)) ∈ (((II ↾t 𝑢) ×t (II ↾t {0})) Cn 𝐶))
221 vex 3192 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑢 ∈ V
222 snex 4874 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {0} ∈ V
223 txrest 21357 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((II ∈ Top ∧ II ∈ Top) ∧ (𝑢 ∈ V ∧ {0} ∈ V)) → ((II ×t II) ↾t (𝑢 × {0})) = ((II ↾t 𝑢) ×t (II ↾t {0})))
22420, 20, 221, 222, 223mp4an 708 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × {0})) = ((II ↾t 𝑢) ×t (II ↾t {0}))
225224oveq1i 6620 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶) = (((II ↾t 𝑢) ×t (II ↾t {0})) Cn 𝐶)
226220, 225syl6eleqr 2709 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
227210, 226eqeltrd 2698 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
228 sneq 4163 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = 0 → {𝑤} = {0})
229228xpeq2d 5104 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → (𝑢 × {𝑤}) = (𝑢 × {0}))
230229reseq2d 5361 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (𝐾 ↾ (𝑢 × {𝑤})) = (𝐾 ↾ (𝑢 × {0})))
231229oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → ((II ×t II) ↾t (𝑢 × {𝑤})) = ((II ×t II) ↾t (𝑢 × {0})))
232231oveq1d 6625 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
233230, 232eleq12d 2692 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 0 → ((𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) ↔ (𝐾 ↾ (𝑢 × {0})) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶)))
234233rspcev 3298 . . . . . . . . . . . . . . . . . . . . . 22 ((0 ∈ 𝑣 ∧ (𝐾 ↾ (𝑢 × {0})) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶)) → ∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
235184, 227, 234syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
236 opelxpi 5113 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑟𝑢 ∧ 0 ∈ 𝑣) → ⟨𝑟, 0⟩ ∈ (𝑢 × 𝑣))
237236adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ (𝑢 × 𝑣))
238 simplrr 800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑣 ∈ II)
239238, 145syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑣 ⊆ (0[,]1))
240 xpss12 5191 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑢 ⊆ (0[,]1) ∧ 𝑣 ⊆ (0[,]1)) → (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1)))
241194, 239, 240syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1)))
24237restuni 20889 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((II ×t II) ∈ Top ∧ (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1))) → (𝑢 × 𝑣) = ((II ×t II) ↾t (𝑢 × 𝑣)))
24322, 241, 242sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) = ((II ×t II) ↾t (𝑢 × 𝑣)))
244237, 243eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((II ×t II) ↾t (𝑢 × 𝑣)))
245 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × 𝑣)) = ((II ×t II) ↾t (𝑢 × 𝑣))
246245cncnpi 21005 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶) ∧ ⟨𝑟, 0⟩ ∈ ((II ×t II) ↾t (𝑢 × 𝑣))) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ ((((II ×t II) ↾t (𝑢 × 𝑣)) CnP 𝐶)‘⟨𝑟, 0⟩))
247246expcom 451 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑟, 0⟩ ∈ ((II ×t II) ↾t (𝑢 × 𝑣)) → ((𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ ((((II ×t II) ↾t (𝑢 × 𝑣)) CnP 𝐶)‘⟨𝑟, 0⟩)))
248244, 247syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ ((((II ×t II) ↾t (𝑢 × 𝑣)) CnP 𝐶)‘⟨𝑟, 0⟩)))
24922a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (II ×t II) ∈ Top)
25020a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → II ∈ Top)
251250, 250, 191, 238, 54syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) ∈ (II ×t II))
252 isopn3i 20809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((II ×t II) ∈ Top ∧ (𝑢 × 𝑣) ∈ (II ×t II)) → ((int‘(II ×t II))‘(𝑢 × 𝑣)) = (𝑢 × 𝑣))
25322, 251, 252sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((int‘(II ×t II))‘(𝑢 × 𝑣)) = (𝑢 × 𝑣))
254237, 253eleqtrrd 2701 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((int‘(II ×t II))‘(𝑢 × 𝑣)))
25537, 1cnprest 21016 . . . . . . . . . . . . . . . . . . . . . . 23 ((((II ×t II) ∈ Top ∧ (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1))) ∧ (⟨𝑟, 0⟩ ∈ ((int‘(II ×t II))‘(𝑢 × 𝑣)) ∧ 𝐾:((0[,]1) × (0[,]1))⟶𝐵)) → (𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩) ↔ (𝐾 ↾ (𝑢 × 𝑣)) ∈ ((((II ×t II) ↾t (𝑢 × 𝑣)) CnP 𝐶)‘⟨𝑟, 0⟩)))
256249, 241, 254, 185, 255syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩) ↔ (𝐾 ↾ (𝑢 × 𝑣)) ∈ ((((II ×t II) ↾t (𝑢 × 𝑣)) CnP 𝐶)‘⟨𝑟, 0⟩)))
257248, 256sylibrd 249 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
258235, 257embantd 59 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
259258expimpd 628 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → (((𝑟𝑢 ∧ 0 ∈ 𝑣) ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
260183, 259syl5bi 232 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → ((𝑟𝑢 ∧ 0 ∈ 𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
261260rexlimdvva 3032 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟 ∈ (0[,]1)) → (∃𝑢 ∈ II ∃𝑣 ∈ II (𝑟𝑢 ∧ 0 ∈ 𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
262182, 261mpd 15 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ (0[,]1)) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩))
263 fveq2 6153 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑟, 0⟩ → (((II ×t II) CnP 𝐶)‘𝑧) = (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩))
264263eleq2d 2684 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑟, 0⟩ → (𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧) ↔ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
265264, 69elrab2 3352 . . . . . . . . . . . . . . . 16 (⟨𝑟, 0⟩ ∈ 𝑀 ↔ (⟨𝑟, 0⟩ ∈ ((0[,]1) × (0[,]1)) ∧ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
266175, 262, 265sylanbrc 697 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (0[,]1)) → ⟨𝑟, 0⟩ ∈ 𝑀)
267 elsni 4170 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {0} → 𝑎 = 0)
268267opeq2d 4382 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ = ⟨𝑟, 0⟩)
269268eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑎 ∈ {0} → (⟨𝑟, 𝑎⟩ ∈ 𝑀 ↔ ⟨𝑟, 0⟩ ∈ 𝑀))
270266, 269syl5ibrcom 237 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (0[,]1)) → (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
271270expimpd 628 . . . . . . . . . . . . 13 (𝜑 → ((𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}) → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
272172, 271syl5bi 232 . . . . . . . . . . . 12 (𝜑 → (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
273171, 272relssdv 5178 . . . . . . . . . . 11 (𝜑 → ((0[,]1) × {0}) ⊆ 𝑀)
274 sneq 4163 . . . . . . . . . . . . . 14 (𝑎 = 0 → {𝑎} = {0})
275274xpeq2d 5104 . . . . . . . . . . . . 13 (𝑎 = 0 → ((0[,]1) × {𝑎}) = ((0[,]1) × {0}))
276275sseq1d 3616 . . . . . . . . . . . 12 (𝑎 = 0 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {0}) ⊆ 𝑀))
277276, 13elrab2 3352 . . . . . . . . . . 11 (0 ∈ 𝐴 ↔ (0 ∈ (0[,]1) ∧ ((0[,]1) × {0}) ⊆ 𝑀))
278169, 273, 277sylanbrc 697 . . . . . . . . . 10 (𝜑 → 0 ∈ 𝐴)
279 ne0i 3902 . . . . . . . . . 10 (0 ∈ 𝐴𝐴 ≠ ∅)
280278, 279syl 17 . . . . . . . . 9 (𝜑𝐴 ≠ ∅)
281 inss2 3817 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ (Clsd‘II)
282281, 166sseldi 3585 . . . . . . . . 9 (𝜑𝐴 ∈ (Clsd‘II))
28314, 16, 167, 280, 282connclo 21141 . . . . . . . 8 (𝜑𝐴 = (0[,]1))
28413, 283syl5reqr 2670 . . . . . . 7 (𝜑 → (0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀})
285 rabid2 3110 . . . . . . 7 ((0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ↔ ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
286284, 285sylib 208 . . . . . 6 (𝜑 → ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
287 iunss 4532 . . . . . 6 ( 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
288286, 287sylibr 224 . . . . 5 (𝜑 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
28912, 288syl5eqss 3633 . . . 4 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ 𝑀)
290289, 69syl6sseq 3635 . . 3 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)})
291 ssrab 3664 . . . 4 (((0[,]1) × (0[,]1)) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)} ↔ (((0[,]1) × (0[,]1)) ⊆ ((0[,]1) × (0[,]1)) ∧ ∀𝑧 ∈ ((0[,]1) × (0[,]1))𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)))
292291simprbi 480 . . 3 (((0[,]1) × (0[,]1)) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)} → ∀𝑧 ∈ ((0[,]1) × (0[,]1))𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧))
293290, 292syl 17 . 2 (𝜑 → ∀𝑧 ∈ ((0[,]1) × (0[,]1))𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧))
294 txtopon 21317 . . . 4 ((II ∈ (TopOn‘(0[,]1)) ∧ II ∈ (TopOn‘(0[,]1))) → (II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1))))
295212, 212, 294mp2an 707 . . 3 (II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1)))
296 cvmtop1 30985 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2972, 296syl 17 . . . 4 (𝜑𝐶 ∈ Top)
2981toptopon 20657 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
299297, 298sylib 208 . . 3 (𝜑𝐶 ∈ (TopOn‘𝐵))
300 cncnp 21007 . . 3 (((II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1))) ∧ 𝐶 ∈ (TopOn‘𝐵)) → (𝐾 ∈ ((II ×t II) Cn 𝐶) ↔ (𝐾:((0[,]1) × (0[,]1))⟶𝐵 ∧ ∀𝑧 ∈ ((0[,]1) × (0[,]1))𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧))))
301295, 299, 300sylancr 694 . 2 (𝜑 → (𝐾 ∈ ((II ×t II) Cn 𝐶) ↔ (𝐾:((0[,]1) × (0[,]1))⟶𝐵 ∧ ∀𝑧 ∈ ((0[,]1) × (0[,]1))𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧))))
3028, 293, 301mpbir2and 956 1 (𝜑𝐾 ∈ ((II ×t II) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153  cop 4159   cuni 4407   ciun 4490  {copab 4677  cmpt 4678   × cxp 5077  ccnv 5078  cres 5081  cima 5082  ccom 5083  Rel wrel 5084   Fn wfn 5847  wf 5848  cfv 5852  crio 6570  (class class class)co 6610  cmpt2 6612  0cc0 9888  1c1 9889  [,]cicc 12128  t crest 16013  Topctop 20630  TopOnctopon 20647  Clsdccld 20743  intcnt 20744  neicnei 20824   Cn ccn 20951   CnP ccnp 20952  Compccmp 21112  Conncconn 21137   ×t ctx 21286  Homeochmeo 21479  IIcii 22601   CovMap ccvm 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966  ax-addf 9967  ax-mulf 9968
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-ec 7696  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-fi 8269  df-sup 8300  df-inf 8301  df-oi 8367  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-q 11741  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-ioo 12129  df-ico 12131  df-icc 12132  df-fz 12277  df-fzo 12415  df-fl 12541  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161  df-sum 14359  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-hom 15898  df-cco 15899  df-rest 16015  df-topn 16016  df-0g 16034  df-gsum 16035  df-topgen 16036  df-pt 16037  df-prds 16040  df-xrs 16094  df-qtop 16099  df-imas 16100  df-xps 16102  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-mulg 17473  df-cntz 17682  df-cmn 18127  df-psmet 19670  df-xmet 19671  df-met 19672  df-bl 19673  df-mopn 19674  df-cnfld 19679  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-cld 20746  df-ntr 20747  df-cls 20748  df-nei 20825  df-cn 20954  df-cnp 20955  df-cmp 21113  df-conn 21138  df-lly 21192  df-nlly 21193  df-tx 21288  df-hmeo 21481  df-xms 22048  df-ms 22049  df-tms 22050  df-ii 22603  df-htpy 22692  df-phtpy 22693  df-phtpc 22714  df-pconn 30946  df-sconn 30947  df-cvm 30981
This theorem is referenced by:  cvmlift2lem13  31040
  Copyright terms: Public domain W3C validator