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Theorem cvmlift2lem12 35319
Description: Lemma for cvmlift2 35321. (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 35312 . 2 (𝜑𝐾:((0[,]1) × (0[,]1))⟶𝐵)
9 iunid 5060 . . . . . . 7 𝑎 ∈ (0[,]1){𝑎} = (0[,]1)
109xpeq2i 5712 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = ((0[,]1) × (0[,]1))
11 xpiundi 5756 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
1210, 11eqtr3i 2767 . . . . 5 ((0[,]1) × (0[,]1)) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
13 iiuni 24907 . . . . . . . . 9 (0[,]1) = II
14 iiconn 24913 . . . . . . . . . 10 II ∈ Conn
1514a1i 11 . . . . . . . . 9 (𝜑 → II ∈ Conn)
16 inss1 4237 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ II
17 iicmp 24912 . . . . . . . . . . . . . . 15 II ∈ Comp
1817a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Comp)
19 iitop 24906 . . . . . . . . . . . . . . 15 II ∈ Top
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Top)
2119, 19txtopi 23598 . . . . . . . . . . . . . . . 16 (II ×t II) ∈ Top
2213neiss2 23109 . . . . . . . . . . . . . . . . . . . . . . . 24 ((II ∈ Top ∧ 𝑢 ∈ ((nei‘II)‘{𝑟})) → {𝑟} ⊆ (0[,]1))
2319, 22mpan 690 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ ((nei‘II)‘{𝑟}) → {𝑟} ⊆ (0[,]1))
24 vex 3484 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑟 ∈ V
2524snss 4785 . . . . . . . . . . . . . . . . . . . . . . 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 3148 . . . . . . . . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))}
3532, 33, 343sstr4i 4035 . . . . . . . . . . . . . . . 16 𝑆 ⊆ ((0[,]1) × (0[,]1))
3619, 19, 13, 13txunii 23601 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = (II ×t II)
3736ntropn 23057 . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . 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 35317 . . . . . . . . . . . . . . . . . . 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 23610 . . . . . . . . . . . . . . . . . . . . . . . 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 4912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑡𝑣𝑣 ∈ II) → 𝑡 II)
5756, 13eleqtrrdi 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((II ∈ Top ∧ 𝑢 ∈ II ∧ 𝑟𝑢) → 𝑢 ∈ ((nei‘II)‘{𝑟}))
6359, 60, 61, 62syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → {𝑐} = {𝑤})
7372xpeq2d 5715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (𝑢 × {𝑐}) = (𝑢 × {𝑤}))
7473reseq2d 5997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (𝐾 ↾ (𝑢 × {𝑐})) = (𝐾 ↾ (𝑢 × {𝑤})))
7573oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → ((II ×t II) ↾t (𝑢 × {𝑐})) = ((II ×t II) ↾t (𝑢 × {𝑤})))
7675oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
7774, 76eleq12d 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3238 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 35318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 35318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3249 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1928 . . . . . . . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)}
9089, 33sseq12i 4014 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))})
91 ssopab2bw 5552 . . . . . . . . . . . . . . . . . . . . . . . . 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 23066 . . . . . . . . . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . . . . . . 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 3484 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
105 opeq2 4874 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑎 → ⟨𝑏, 𝑤⟩ = ⟨𝑏, 𝑎⟩)
106105eleq1d 2826 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑎 → (⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆)))
107104, 106ralsn 4681 . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (0[,]1)) → ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
111 dfss3 3972 . . . . . . . . . . . . . . . 16 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑢 ∈ ((0[,]1) × {𝑎})𝑢 ∈ ((int‘(II ×t II))‘𝑆))
112 eleq1 2829 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑏, 𝑤⟩ → (𝑢 ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆)))
113112ralxp 5852 . . . . . . . . . . . . . . . 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 23648 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (0[,]1)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)))
11836ntrss2 23065 . . . . . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . . . 19 ((0[,]1) × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)}
123122, 33sseq12i 4014 . . . . . . . . . . . . . . . . . 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 5552 . . . . . . . . . . . . . . . . . . 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 3195 . . . . . . . . . . . . . . . . . . 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 3289 . . . . . . . . . . . . . . . . . . 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 3083 . . . . . . . . . . . . . . . . . . 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 35307 . . . . . . . . . . . . . . . . . . . 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 3094 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
135134ralbii 3093 . . . . . . . . . . . . . . . . . . . . 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 35307 . . . . . . . . . . . . . . . . . . . . 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 3460 . . . . . . . . . . . . . . . . . . . . 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 4937 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ II → 𝑣 II)
145144, 13sseqtrrdi 4025 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ II → 𝑣 ⊆ (0[,]1))
146145adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → 𝑣 ⊆ (0[,]1))
147146sselda 3983 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → 𝑡 ∈ (0[,]1))
148 sneq 4636 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑡 → {𝑎} = {𝑡})
149148xpeq2d 5715 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑡 → ((0[,]1) × {𝑎}) = ((0[,]1) × {𝑡}))
150149sseq1d 4015 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑡 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
151150, 140elrab2 3695 . . . . . . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . . 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 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
162 ssrab2 4080 . . . . . . . . . . . . 13 {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ⊆ (0[,]1)
163140, 162eqsstri 4030 . . . . . . . . . . . 12 𝐴 ⊆ (0[,]1)
16413isclo 23095 . . . . . . . . . . . 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 3981 . . . . . . . . 9 (𝜑𝐴 ∈ II)
168 0elunit 13509 . . . . . . . . . . . 12 0 ∈ (0[,]1)
169168a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ∈ (0[,]1))
170 relxp 5703 . . . . . . . . . . . . 13 Rel ((0[,]1) × {0})
171170a1i 11 . . . . . . . . . . . 12 (𝜑 → Rel ((0[,]1) × {0}))
172 opelxp 5721 . . . . . . . . . . . . 13 (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) ↔ (𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}))
173 id 22 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ (0[,]1) → 𝑟 ∈ (0[,]1))
174 opelxpi 5722 . . . . . . . . . . . . . . . . 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 35317 . . . . . . . . . . . . . . . . 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 1089 . . . . . . . . . . . . . . . . . . 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 7564 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5996 . . . . . . . . . . . . . . . . . . . . . . . 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 4937 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 ∈ II → 𝑢 II)
192191, 13sseqtrrdi 4025 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 ∈ II → 𝑢 ⊆ (0[,]1))
193190, 192syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ⊆ (0[,]1))
194169snssd 4809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → {0} ⊆ (0[,]1))
195194ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → {0} ⊆ (0[,]1))
196 resmpo 7553 . . . . . . . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 35315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 ∈ {0} → 𝑤 = 0)
204203oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {0} → (𝑏𝐾𝑤) = (𝑏𝐾0))
205204eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢𝑤 ∈ {0}) → (𝑏𝐾𝑤) = (𝐻𝑏))
208207mpoeq3dva 7510 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
209189, 197, 2083eqtrd 2781 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
210 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t 𝑢) = (II ↾t 𝑢)
211 iitopon 24905 . . . . . . . . . . . . . . . . . . . . . . . . . 26 II ∈ (TopOn‘(0[,]1))
212211a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → II ∈ (TopOn‘(0[,]1)))
213 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t {0}) = (II ↾t {0})
214212, 212cnmpt1st 23676 . . . . . . . . . . . . . . . . . . . . . . . . . 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 35309 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
216215simp1d 1143 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23680 . . . . . . . . . . . . . . . . . . . . . . . . 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 23685 . . . . . . . . . . . . . . . . . . . . . . . 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 3484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑢 ∈ V
221 snex 5436 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {0} ∈ V
222 txrest 23639 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶) = (((II ↾t 𝑢) ×t (II ↾t {0})) Cn 𝐶)
225219, 224eleqtrrdi 2852 . . . . . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
227 sneq 4636 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = 0 → {𝑤} = {0})
228227xpeq2d 5715 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → (𝑢 × {𝑤}) = (𝑢 × {0}))
229228reseq2d 5997 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (𝐾 ↾ (𝑢 × {𝑤})) = (𝐾 ↾ (𝑢 × {0})))
230228oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → ((II ×t II) ↾t (𝑢 × {𝑤})) = ((II ×t II) ↾t (𝑢 × {0})))
231230oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
232229, 231eleq12d 2835 . . . . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . . . . . . . . . . . . . . . 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 23170 . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((II ×t II) ↾t (𝑢 × 𝑣)))
244 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × 𝑣)) = ((II ×t II) ↾t (𝑢 × 𝑣))
245244cncnpi 23286 . . . . . . . . . . . . . . . . . . . . . . . 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 23090 . . . . . . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((int‘(II ×t II))‘(𝑢 × 𝑣)))
25436, 1cnprest 23297 . . . . . . . . . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑟, 0⟩ → (𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧) ↔ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
264263, 68elrab2 3695 . . . . . . . . . . . . . . . 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 4643 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {0} → 𝑎 = 0)
267266opeq2d 4880 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ = ⟨𝑟, 0⟩)
268267eleq1d 2826 . . . . . . . . . . . . . . 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 5798 . . . . . . . . . . 11 (𝜑 → ((0[,]1) × {0}) ⊆ 𝑀)
273 sneq 4636 . . . . . . . . . . . . . 14 (𝑎 = 0 → {𝑎} = {0})
274273xpeq2d 5715 . . . . . . . . . . . . 13 (𝑎 = 0 → ((0[,]1) × {𝑎}) = ((0[,]1) × {0}))
275274sseq1d 4015 . . . . . . . . . . . 12 (𝑎 = 0 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {0}) ⊆ 𝑀))
276275, 140elrab2 3695 . . . . . . . . . . 11 (0 ∈ 𝐴 ↔ (0 ∈ (0[,]1) ∧ ((0[,]1) × {0}) ⊆ 𝑀))
277169, 272, 276sylanbrc 583 . . . . . . . . . 10 (𝜑 → 0 ∈ 𝐴)
278277ne0d 4342 . . . . . . . . 9 (𝜑𝐴 ≠ ∅)
279 inss2 4238 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ (Clsd‘II)
280279, 166sselid 3981 . . . . . . . . 9 (𝜑𝐴 ∈ (Clsd‘II))
28113, 15, 167, 278, 280connclo 23423 . . . . . . . 8 (𝜑𝐴 = (0[,]1))
282281, 140eqtr3di 2792 . . . . . . 7 (𝜑 → (0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀})
283 rabid2 3470 . . . . . . 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 5045 . . . . . 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 4022 . . . 4 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ 𝑀)
288287, 68sseqtrdi 4024 . . 3 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)})
289 ssrab 4073 . . . 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 23599 . . . 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 35265 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2952, 294syl 17 . . . 4 (𝜑𝐶 ∈ Top)
2961toptopon 22923 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
297295, 296sylib 218 . . 3 (𝜑𝐶 ∈ (TopOn‘𝐵))
298 cncnp 23288 . . 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 1087  wal 1538   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  cop 4632   cuni 4907   ciun 4991  {copab 5205  cmpt 5225   × cxp 5683  ccnv 5684  cres 5687  cima 5688  ccom 5689  Rel wrel 5690   Fn wfn 6556  wf 6557  cfv 6561  crio 7387  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156  [,]cicc 13390  t crest 17465  Topctop 22899  TopOnctopon 22916  Clsdccld 23024  intcnt 23025  neicnei 23105   Cn ccn 23232   CnP ccnp 23233  Compccmp 23394  Conncconn 23419   ×t ctx 23568  Homeochmeo 23761  IIcii 24901   CovMap ccvm 35260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cn 23235  df-cnp 23236  df-cmp 23395  df-conn 23420  df-lly 23474  df-nlly 23475  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-ii 24903  df-cncf 24904  df-htpy 25002  df-phtpy 25003  df-phtpc 25024  df-pconn 35226  df-sconn 35227  df-cvm 35261
This theorem is referenced by:  cvmlift2lem13  35320
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