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Theorem cvmlift2lem12 35298
Description: Lemma for cvmlift2 35300. (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 35291 . 2 (𝜑𝐾:((0[,]1) × (0[,]1))⟶𝐵)
9 iunid 5064 . . . . . . 7 𝑎 ∈ (0[,]1){𝑎} = (0[,]1)
109xpeq2i 5715 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = ((0[,]1) × (0[,]1))
11 xpiundi 5758 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
1210, 11eqtr3i 2764 . . . . 5 ((0[,]1) × (0[,]1)) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
13 iiuni 24920 . . . . . . . . 9 (0[,]1) = II
14 iiconn 24926 . . . . . . . . . 10 II ∈ Conn
1514a1i 11 . . . . . . . . 9 (𝜑 → II ∈ Conn)
16 inss1 4244 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ II
17 iicmp 24925 . . . . . . . . . . . . . . 15 II ∈ Comp
1817a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Comp)
19 iitop 24919 . . . . . . . . . . . . . . 15 II ∈ Top
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Top)
2119, 19txtopi 23613 . . . . . . . . . . . . . . . 16 (II ×t II) ∈ Top
2213neiss2 23124 . . . . . . . . . . . . . . . . . . . . . . . 24 ((II ∈ Top ∧ 𝑢 ∈ ((nei‘II)‘{𝑟})) → {𝑟} ⊆ (0[,]1))
2319, 22mpan 690 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ ((nei‘II)‘{𝑟}) → {𝑟} ⊆ (0[,]1))
24 vex 3481 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑟 ∈ V
2524snss 4789 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ (0[,]1) ↔ {𝑟} ⊆ (0[,]1))
2623, 25sylibr 234 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 ∈ ((nei‘II)‘{𝑟}) → 𝑟 ∈ (0[,]1))
2726a1d 25 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ ((nei‘II)‘{𝑟}) → (((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → 𝑟 ∈ (0[,]1)))
2827rexlimiv 3145 . . . . . . . . . . . . . . . . . . . 20 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → 𝑟 ∈ (0[,]1))
2928adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → 𝑟 ∈ (0[,]1))
30 simpl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → 𝑡 ∈ (0[,]1))
3129, 30jca 511 . . . . . . . . . . . . . . . . . 18 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1)))
3231ssopab2i 5559 . . . . . . . . . . . . . . . . 17 {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))}
33 cvmlift2.s . . . . . . . . . . . . . . . . 17 𝑆 = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))}
34 df-xp 5694 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))}
3532, 33, 343sstr4i 4038 . . . . . . . . . . . . . . . 16 𝑆 ⊆ ((0[,]1) × (0[,]1))
3619, 19, 13, 13txunii 23616 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = (II ×t II)
3736ntropn 23072 . . . . . . . . . . . . . . . 16 (((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) → ((int‘(II ×t II))‘𝑆) ∈ (II ×t II))
3821, 35, 37mp2an 692 . . . . . . . . . . . . . . 15 ((int‘(II ×t II))‘𝑆) ∈ (II ×t II)
3938a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → ((int‘(II ×t II))‘𝑆) ∈ (II ×t II))
402adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
413adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
424adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑃𝐵)
435adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → (𝐹𝑃) = (0𝐺0))
44 eqid 2734 . . . . . . . . . . . . . . . . . . . 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))))})
45 simprr 773 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑏 ∈ (0[,]1))
46 simprl 771 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑎 ∈ (0[,]1))
471, 40, 41, 42, 43, 6, 7, 44, 45, 46cvmlift2lem10 35296 . . . . . . . . . . . . . . . . . . 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))))
4821a1i 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)
4935a1i 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)))
5019a1i 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)
51 simplrl 777 . . . . . . . . . . . . . . . . . . . . . . . 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)
52 simplrr 778 . . . . . . . . . . . . . . . . . . . . . . . 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)
53 txopn 23625 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ∈ Top ∧ II ∈ Top) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → (𝑢 × 𝑣) ∈ (II ×t II))
5450, 50, 51, 52, 53syl22anc 839 . . . . . . . . . . . . . . . . . . . . . . 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))
55 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑟𝑢𝑡𝑣) → 𝑡𝑣)
56 elunii 4916 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑡𝑣𝑣 ∈ II) → 𝑡 II)
5756, 13eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑡𝑣𝑣 ∈ II) → 𝑡 ∈ (0[,]1))
5855, 52, 57syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
5919a1i 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)
6051adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
61 simprl 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
62 opnneip 23142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((II ∈ Top ∧ 𝑢 ∈ II ∧ 𝑟𝑢) → 𝑢 ∈ ((nei‘II)‘{𝑟}))
6359, 60, 61, 62syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))
6440ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
6541ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
6642ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
6743ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
68 cvmlift2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}
6952adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
70 simplr2 1215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
71 simprr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
72 sneq 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → {𝑐} = {𝑤})
7372xpeq2d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (𝑢 × {𝑐}) = (𝑢 × {𝑤}))
7473reseq2d 5999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (𝐾 ↾ (𝑢 × {𝑐})) = (𝐾 ↾ (𝑢 × {𝑤})))
7573oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → ((II ×t II) ↾t (𝑢 × {𝑐})) = ((II ×t II) ↾t (𝑢 × {𝑤})))
7675oveq1d 7445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
7774, 76eleq12d 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
7877cbvrexvw 3235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
79 simplr3 1216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
8078, 79biimtrid 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
811, 64, 65, 66, 67, 6, 7, 68, 60, 69, 70, 71, 80cvmlift2lem11 35297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
821, 64, 65, 66, 67, 6, 7, 68, 60, 69, 71, 70, 80cvmlift2lem11 35297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
8381, 82impbid 212 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
84 rspe 3246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑢 ∈ ((nei‘II)‘{𝑟}) ∧ ((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
8563, 83, 84syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
8658, 85jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
8786ex 412 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
8887alrimivv 1925 . . . . . . . . . . . . . . . . . . . . . . . 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))))
89 df-xp 5694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)}
9089, 33sseq12i 4025 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))})
91 ssopab2bw 5556 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
9290, 91bitri 275 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ ∀𝑟𝑡((𝑟𝑢𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
9388, 92sylibr 234 . . . . . . . . . . . . . . . . . . . . . . 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)
9436ssntr 23081 . . . . . . . . . . . . . . . . . . . . . . 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))
9548, 49, 54, 93, 94syl22anc 839 . . . . . . . . . . . . . . . . . . . . . 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))
96 simpr1 1193 . . . . . . . . . . . . . . . . . . . . . . 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)
97 simpr2 1194 . . . . . . . . . . . . . . . . . . . . . . 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)
98 opelxpi 5725 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑢𝑎𝑣) → ⟨𝑏, 𝑎⟩ ∈ (𝑢 × 𝑣))
9996, 97, 98syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 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))
10095, 99sseldd 3995 . . . . . . . . . . . . . . . . . . . . 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))
101100ex 412 . . . . . . . . . . . . . . . . . . . 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)))
102101rexlimdvva 3210 . . . . . . . . . . . . . . . . . . 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)))
10347, 102mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆))
104 vex 3481 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
105 opeq2 4878 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑎 → ⟨𝑏, 𝑤⟩ = ⟨𝑏, 𝑎⟩)
106105eleq1d 2823 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑎 → (⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆)))
107104, 106ralsn 4685 . . . . . . . . . . . . . . . . . 18 (∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆))
108103, 107sylibr 234 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → ∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
109108anassrs 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑏 ∈ (0[,]1)) → ∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
110109ralrimiva 3143 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (0[,]1)) → ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
111 dfss3 3983 . . . . . . . . . . . . . . . 16 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑢 ∈ ((0[,]1) × {𝑎})𝑢 ∈ ((int‘(II ×t II))‘𝑆))
112 eleq1 2826 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑏, 𝑤⟩ → (𝑢 ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆)))
113112ralxp 5854 . . . . . . . . . . . . . . . 16 (∀𝑢 ∈ ((0[,]1) × {𝑎})𝑢 ∈ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
114111, 113bitri 275 . . . . . . . . . . . . . . 15 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
115110, 114sylibr 234 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → ((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆))
116 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → 𝑎 ∈ (0[,]1))
11713, 13, 18, 20, 39, 115, 116txtube 23663 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (0[,]1)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)))
11836ntrss2 23080 . . . . . . . . . . . . . . . . . . 19 (((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) → ((int‘(II ×t II))‘𝑆) ⊆ 𝑆)
11921, 35, 118mp2an 692 . . . . . . . . . . . . . . . . . 18 ((int‘(II ×t II))‘𝑆) ⊆ 𝑆
120 sstr 4003 . . . . . . . . . . . . . . . . . 18 ((((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) ∧ ((int‘(II ×t II))‘𝑆) ⊆ 𝑆) → ((0[,]1) × 𝑣) ⊆ 𝑆)
121119, 120mpan2 691 . . . . . . . . . . . . . . . . 17 (((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) → ((0[,]1) × 𝑣) ⊆ 𝑆)
122 df-xp 5694 . . . . . . . . . . . . . . . . . . 19 ((0[,]1) × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)}
123122, 33sseq12i 4025 . . . . . . . . . . . . . . . . . 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))})
124 ssopab2bw 5556 . . . . . . . . . . . . . . . . . . 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))))
125 r2al 3192 . . . . . . . . . . . . . . . . . . 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))))
126 ralcom 3286 . . . . . . . . . . . . . . . . . . 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)))
127124, 125, 1263bitr2i 299 . . . . . . . . . . . . . . . . . 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)))
128123, 127bitri 275 . . . . . . . . . . . . . . . . 17 (((0[,]1) × 𝑣) ⊆ 𝑆 ↔ ∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
129121, 128sylib 218 . . . . . . . . . . . . . . . 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)))
130 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))
131130ralimi 3080 . . . . . . . . . . . . . . . . . . 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))
132 cvmlift2lem1 35286 . . . . . . . . . . . . . . . . . . . 20 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑎}) ⊆ 𝑀 → ((0[,]1) × {𝑡}) ⊆ 𝑀))
133 bicom 222 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
134133rexbii 3091 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
135134ralbii 3090 . . . . . . . . . . . . . . . . . . . . 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))
136 cvmlift2lem1 35286 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀) → (((0[,]1) × {𝑡}) ⊆ 𝑀 → ((0[,]1) × {𝑎}) ⊆ 𝑀))
137135, 136sylbi 217 . . . . . . . . . . . . . . . . . . . 20 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑡}) ⊆ 𝑀 → ((0[,]1) × {𝑎}) ⊆ 𝑀))
138132, 137impbid 212 . . . . . . . . . . . . . . . . . . 19 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
139131, 138syl 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))
140 cvmlift2.a . . . . . . . . . . . . . . . . . . . . . 22 𝐴 = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀}
141140reqabi 3456 . . . . . . . . . . . . . . . . . . . . 21 (𝑎𝐴 ↔ (𝑎 ∈ (0[,]1) ∧ ((0[,]1) × {𝑎}) ⊆ 𝑀))
142141baib 535 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (0[,]1) → (𝑎𝐴 ↔ ((0[,]1) × {𝑎}) ⊆ 𝑀))
143142ad3antlr 731 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → (𝑎𝐴 ↔ ((0[,]1) × {𝑎}) ⊆ 𝑀))
144 elssuni 4941 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ II → 𝑣 II)
145144, 13sseqtrrdi 4046 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ II → 𝑣 ⊆ (0[,]1))
146145adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → 𝑣 ⊆ (0[,]1))
147146sselda 3994 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → 𝑡 ∈ (0[,]1))
148 sneq 4640 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑡 → {𝑎} = {𝑡})
149148xpeq2d 5718 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑡 → ((0[,]1) × {𝑎}) = ((0[,]1) × {𝑡}))
150149sseq1d 4026 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑡 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
151150, 140elrab2 3697 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐴 ↔ (𝑡 ∈ (0[,]1) ∧ ((0[,]1) × {𝑡}) ⊆ 𝑀))
152151baib 535 . . . . . . . . . . . . . . . . . . . 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 345 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → ((𝑎𝐴𝑡𝐴) ↔ (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀)))
155139, 154imbitrrid 246 . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . 16 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → (∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
157129, 156syl5 34 . . . . . . . . . . . . . . 15 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → (((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) → ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
158157anim2d 612 . . . . . . . . . . . . . 14 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → ((𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)) → (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
159158reximdva 3165 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (0[,]1)) → (∃𝑣 ∈ II (𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
160117, 159mpd 15 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (0[,]1)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
161160ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
162 ssrab2 4089 . . . . . . . . . . . . 13 {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ⊆ (0[,]1)
163140, 162eqsstri 4029 . . . . . . . . . . . 12 𝐴 ⊆ (0[,]1)
16413isclo 23110 . . . . . . . . . . . 12 ((II ∈ Top ∧ 𝐴 ⊆ (0[,]1)) → (𝐴 ∈ (II ∩ (Clsd‘II)) ↔ ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
16519, 163, 164mp2an 692 . . . . . . . . . . 11 (𝐴 ∈ (II ∩ (Clsd‘II)) ↔ ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
166161, 165sylibr 234 . . . . . . . . . 10 (𝜑𝐴 ∈ (II ∩ (Clsd‘II)))
16716, 166sselid 3992 . . . . . . . . 9 (𝜑𝐴 ∈ II)
168 0elunit 13505 . . . . . . . . . . . 12 0 ∈ (0[,]1)
169168a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ∈ (0[,]1))
170 relxp 5706 . . . . . . . . . . . . 13 Rel ((0[,]1) × {0})
171170a1i 11 . . . . . . . . . . . 12 (𝜑 → Rel ((0[,]1) × {0}))
172 opelxp 5724 . . . . . . . . . . . . 13 (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) ↔ (𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}))
173 id 22 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ (0[,]1) → 𝑟 ∈ (0[,]1))
174 opelxpi 5725 . . . . . . . . . . . . . . . . 17 ((𝑟 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨𝑟, 0⟩ ∈ ((0[,]1) × (0[,]1)))
175173, 169, 174syl2anr 597 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ (0[,]1)) → ⟨𝑟, 0⟩ ∈ ((0[,]1) × (0[,]1)))
1762adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
1773adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
1784adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝑃𝐵)
1795adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → (𝐹𝑃) = (0𝐺0))
180 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝑟 ∈ (0[,]1))
181168a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 0 ∈ (0[,]1))
1821, 176, 177, 178, 179, 6, 7, 44, 180, 181cvmlift2lem10 35296 . . . . . . . . . . . . . . . . 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 1088 . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 0 ∈ 𝑣)
1858ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
186185ffnd 6737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾 Fn ((0[,]1) × (0[,]1)))
187 fnov 7563 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐾 Fn ((0[,]1) × (0[,]1)) ↔ 𝐾 = (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)))
188186, 187sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾 = (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)))
189188reseq1d 5998 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) = ((𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)) ↾ (𝑢 × {0})))
190 simplrl 777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ∈ II)
191 elssuni 4941 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 ∈ II → 𝑢 II)
192191, 13sseqtrrdi 4046 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 ∈ II → 𝑢 ⊆ (0[,]1))
193190, 192syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ⊆ (0[,]1))
194169snssd 4813 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → {0} ⊆ (0[,]1))
195194ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → {0} ⊆ (0[,]1))
196 resmpo 7552 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 ⊆ (0[,]1) ∧ {0} ⊆ (0[,]1)) → ((𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)) ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)))
197193, 195, 196syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)) ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)))
198193sselda 3994 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → 𝑏 ∈ (0[,]1))
199 simplll 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝜑)
2001, 2, 3, 4, 5, 6, 7cvmlift2lem8 35294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑏 ∈ (0[,]1)) → (𝑏𝐾0) = (𝐻𝑏))
201199, 200sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏 ∈ (0[,]1)) → (𝑏𝐾0) = (𝐻𝑏))
202198, 201syldan 591 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → (𝑏𝐾0) = (𝐻𝑏))
203 elsni 4647 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 ∈ {0} → 𝑤 = 0)
204203oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {0} → (𝑏𝐾𝑤) = (𝑏𝐾0))
205204eqeq1d 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ {0} → ((𝑏𝐾𝑤) = (𝐻𝑏) ↔ (𝑏𝐾0) = (𝐻𝑏)))
206202, 205syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → (𝑤 ∈ {0} → (𝑏𝐾𝑤) = (𝐻𝑏)))
2072063impia 1116 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢𝑤 ∈ {0}) → (𝑏𝐾𝑤) = (𝐻𝑏))
208207mpoeq3dva 7509 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
209189, 197, 2083eqtrd 2778 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
210 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t 𝑢) = (II ↾t 𝑢)
211 iitopon 24918 . . . . . . . . . . . . . . . . . . . . . . . . . 26 II ∈ (TopOn‘(0[,]1))
212211a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → II ∈ (TopOn‘(0[,]1)))
213 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t {0}) = (II ↾t {0})
214212, 212cnmpt1st 23691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ 𝑏) ∈ ((II ×t II) Cn II))
2151, 2, 3, 4, 5, 6cvmlift2lem2 35288 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
216215simp1d 1141 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐻 ∈ (II Cn 𝐶))
217199, 216syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐻 ∈ (II Cn 𝐶))
218212, 212, 214, 217cnmpt21f 23695 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝐻𝑏)) ∈ ((II ×t II) Cn 𝐶))
219210, 212, 193, 213, 212, 195, 218cnmpt2res 23700 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)) ∈ (((II ↾t 𝑢) ×t (II ↾t {0})) Cn 𝐶))
220 vex 3481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑢 ∈ V
221 snex 5441 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {0} ∈ V
222 txrest 23654 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((II ∈ Top ∧ II ∈ Top) ∧ (𝑢 ∈ V ∧ {0} ∈ V)) → ((II ×t II) ↾t (𝑢 × {0})) = ((II ↾t 𝑢) ×t (II ↾t {0})))
22319, 19, 220, 221, 222mp4an 693 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × {0})) = ((II ↾t 𝑢) ×t (II ↾t {0}))
224223oveq1i 7440 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶) = (((II ↾t 𝑢) ×t (II ↾t {0})) Cn 𝐶)
225219, 224eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
226209, 225eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
227 sneq 4640 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = 0 → {𝑤} = {0})
228227xpeq2d 5718 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → (𝑢 × {𝑤}) = (𝑢 × {0}))
229228reseq2d 5999 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (𝐾 ↾ (𝑢 × {𝑤})) = (𝐾 ↾ (𝑢 × {0})))
230228oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → ((II ×t II) ↾t (𝑢 × {𝑤})) = ((II ×t II) ↾t (𝑢 × {0})))
231230oveq1d 7445 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
232229, 231eleq12d 2832 . . . . . . . . . . . . . . . . . . . . . . 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)))
233232rspcev 3621 . . . . . . . . . . . . . . . . . . . . . 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))
234184, 226, 233syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
235 opelxpi 5725 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑟𝑢 ∧ 0 ∈ 𝑣) → ⟨𝑟, 0⟩ ∈ (𝑢 × 𝑣))
236235adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ (𝑢 × 𝑣))
237 simplrr 778 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑣 ∈ II)
238237, 145syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑣 ⊆ (0[,]1))
239 xpss12 5703 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑢 ⊆ (0[,]1) ∧ 𝑣 ⊆ (0[,]1)) → (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1)))
240193, 238, 239syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1)))
24136restuni 23185 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((II ×t II) ∈ Top ∧ (𝑢 × 𝑣) ⊆ ((0[,]1) × (0[,]1))) → (𝑢 × 𝑣) = ((II ×t II) ↾t (𝑢 × 𝑣)))
24221, 240, 241sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) = ((II ×t II) ↾t (𝑢 × 𝑣)))
243236, 242eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((II ×t II) ↾t (𝑢 × 𝑣)))
244 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × 𝑣)) = ((II ×t II) ↾t (𝑢 × 𝑣))
245244cncnpi 23301 . . . . . . . . . . . . . . . . . . . . . . . 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⟩))
246245expcom 413 . . . . . . . . . . . . . . . . . . . . . . 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⟩)))
247243, 246syl 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⟩)))
24821a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (II ×t II) ∈ Top)
24919a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → II ∈ Top)
250249, 249, 190, 237, 53syl22anc 839 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) ∈ (II ×t II))
251 isopn3i 23105 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((II ×t II) ∈ Top ∧ (𝑢 × 𝑣) ∈ (II ×t II)) → ((int‘(II ×t II))‘(𝑢 × 𝑣)) = (𝑢 × 𝑣))
25221, 250, 251sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ((int‘(II ×t II))‘(𝑢 × 𝑣)) = (𝑢 × 𝑣))
253236, 252eleqtrrd 2841 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((int‘(II ×t II))‘(𝑢 × 𝑣)))
25436, 1cnprest 23312 . . . . . . . . . . . . . . . . . . . . . . 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⟩)))
255248, 240, 253, 185, 254syl22anc 839 . . . . . . . . . . . . . . . . . . . . . 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⟩)))
256247, 255sylibrd 259 . . . . . . . . . . . . . . . . . . . . 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⟩)))
257234, 256embantd 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⟩)))
258257expimpd 453 . . . . . . . . . . . . . . . . . . 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⟩)))
259183, 258biimtrid 242 . . . . . . . . . . . . . . . . . 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⟩)))
260259rexlimdvva 3210 . . . . . . . . . . . . . . . . 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⟩)))
261182, 260mpd 15 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ (0[,]1)) → 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩))
262 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑟, 0⟩ → (((II ×t II) CnP 𝐶)‘𝑧) = (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩))
263262eleq2d 2824 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑟, 0⟩ → (𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧) ↔ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
264263, 68elrab2 3697 . . . . . . . . . . . . . . . 16 (⟨𝑟, 0⟩ ∈ 𝑀 ↔ (⟨𝑟, 0⟩ ∈ ((0[,]1) × (0[,]1)) ∧ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
265175, 261, 264sylanbrc 583 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (0[,]1)) → ⟨𝑟, 0⟩ ∈ 𝑀)
266 elsni 4647 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {0} → 𝑎 = 0)
267266opeq2d 4884 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ = ⟨𝑟, 0⟩)
268267eleq1d 2823 . . . . . . . . . . . . . . 15 (𝑎 ∈ {0} → (⟨𝑟, 𝑎⟩ ∈ 𝑀 ↔ ⟨𝑟, 0⟩ ∈ 𝑀))
269265, 268syl5ibrcom 247 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (0[,]1)) → (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
270269expimpd 453 . . . . . . . . . . . . 13 (𝜑 → ((𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}) → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
271172, 270biimtrid 242 . . . . . . . . . . . 12 (𝜑 → (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
272171, 271relssdv 5800 . . . . . . . . . . 11 (𝜑 → ((0[,]1) × {0}) ⊆ 𝑀)
273 sneq 4640 . . . . . . . . . . . . . 14 (𝑎 = 0 → {𝑎} = {0})
274273xpeq2d 5718 . . . . . . . . . . . . 13 (𝑎 = 0 → ((0[,]1) × {𝑎}) = ((0[,]1) × {0}))
275274sseq1d 4026 . . . . . . . . . . . 12 (𝑎 = 0 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {0}) ⊆ 𝑀))
276275, 140elrab2 3697 . . . . . . . . . . 11 (0 ∈ 𝐴 ↔ (0 ∈ (0[,]1) ∧ ((0[,]1) × {0}) ⊆ 𝑀))
277169, 272, 276sylanbrc 583 . . . . . . . . . 10 (𝜑 → 0 ∈ 𝐴)
278277ne0d 4347 . . . . . . . . 9 (𝜑𝐴 ≠ ∅)
279 inss2 4245 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ (Clsd‘II)
280279, 166sselid 3992 . . . . . . . . 9 (𝜑𝐴 ∈ (Clsd‘II))
28113, 15, 167, 278, 280connclo 23438 . . . . . . . 8 (𝜑𝐴 = (0[,]1))
282281, 140eqtr3di 2789 . . . . . . 7 (𝜑 → (0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀})
283 rabid2 3467 . . . . . . 7 ((0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ↔ ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
284282, 283sylib 218 . . . . . 6 (𝜑 → ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
285 iunss 5049 . . . . . 6 ( 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
286284, 285sylibr 234 . . . . 5 (𝜑 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
28712, 286eqsstrid 4043 . . . 4 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ 𝑀)
288287, 68sseqtrdi 4045 . . 3 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)})
289 ssrab 4082 . . . 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)))
290289simprbi 496 . . 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))
291288, 290syl 17 . 2 (𝜑 → ∀𝑧 ∈ ((0[,]1) × (0[,]1))𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧))
292 txtopon 23614 . . . 4 ((II ∈ (TopOn‘(0[,]1)) ∧ II ∈ (TopOn‘(0[,]1))) → (II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1))))
293211, 211, 292mp2an 692 . . 3 (II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1)))
294 cvmtop1 35244 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2952, 294syl 17 . . . 4 (𝜑𝐶 ∈ Top)
2961toptopon 22938 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
297295, 296sylib 218 . . 3 (𝜑𝐶 ∈ (TopOn‘𝐵))
298 cncnp 23303 . . 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))))
299293, 297, 298sylancr 587 . 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))))
3008, 291, 299mpbir2and 713 1 (𝜑𝐾 ∈ ((II ×t II) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1534   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cdif 3959  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630  cop 4636   cuni 4911   ciun 4995  {copab 5209  cmpt 5230   × cxp 5686  ccnv 5687  cres 5690  cima 5691  ccom 5692  Rel wrel 5693   Fn wfn 6557  wf 6558  cfv 6562  crio 7386  (class class class)co 7430  cmpo 7432  0cc0 11152  1c1 11153  [,]cicc 13386  t crest 17466  Topctop 22914  TopOnctopon 22931  Clsdccld 23039  intcnt 23040  neicnei 23120   Cn ccn 23247   CnP ccnp 23248  Compccmp 23409  Conncconn 23434   ×t ctx 23583  Homeochmeo 23776  IIcii 24914   CovMap ccvm 35239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-ec 8745  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-cn 23250  df-cnp 23251  df-cmp 23410  df-conn 23435  df-lly 23489  df-nlly 23490  df-tx 23585  df-hmeo 23778  df-xms 24345  df-ms 24346  df-tms 24347  df-ii 24916  df-cncf 24917  df-htpy 25015  df-phtpy 25016  df-phtpc 25037  df-pconn 35205  df-sconn 35206  df-cvm 35240
This theorem is referenced by:  cvmlift2lem13  35299
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