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Theorem cvmlift2lem12 33411
Description: Lemma for cvmlift2 33413. (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 33404 . 2 (𝜑𝐾:((0[,]1) × (0[,]1))⟶𝐵)
9 iunid 5002 . . . . . . 7 𝑎 ∈ (0[,]1){𝑎} = (0[,]1)
109xpeq2i 5634 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = ((0[,]1) × (0[,]1))
11 xpiundi 5675 . . . . . 6 ((0[,]1) × 𝑎 ∈ (0[,]1){𝑎}) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
1210, 11eqtr3i 2766 . . . . 5 ((0[,]1) × (0[,]1)) = 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎})
13 iiuni 24124 . . . . . . . . 9 (0[,]1) = II
14 iiconn 24130 . . . . . . . . . 10 II ∈ Conn
1514a1i 11 . . . . . . . . 9 (𝜑 → II ∈ Conn)
16 inss1 4172 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ II
17 iicmp 24129 . . . . . . . . . . . . . . 15 II ∈ Comp
1817a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Comp)
19 iitop 24123 . . . . . . . . . . . . . . 15 II ∈ Top
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → II ∈ Top)
2119, 19txtopi 22821 . . . . . . . . . . . . . . . 16 (II ×t II) ∈ Top
2213neiss2 22332 . . . . . . . . . . . . . . . . . . . . . . . 24 ((II ∈ Top ∧ 𝑢 ∈ ((nei‘II)‘{𝑟})) → {𝑟} ⊆ (0[,]1))
2319, 22mpan 687 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ ((nei‘II)‘{𝑟}) → {𝑟} ⊆ (0[,]1))
24 vex 3444 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑟 ∈ V
2524snss 4730 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ (0[,]1) ↔ {𝑟} ⊆ (0[,]1))
2623, 25sylibr 233 . . . . . . . . . . . . . . . . . . . . . 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 3141 . . . . . . . . . . . . . . . . . . . 20 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → 𝑟 ∈ (0[,]1))
2928adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → 𝑟 ∈ (0[,]1))
30 simpl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → 𝑡 ∈ (0[,]1))
3129, 30jca 512 . . . . . . . . . . . . . . . . . 18 ((𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)) → (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1)))
3231ssopab2i 5482 . . . . . . . . . . . . . . . . 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 5613 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))}
3532, 33, 343sstr4i 3973 . . . . . . . . . . . . . . . 16 𝑆 ⊆ ((0[,]1) × (0[,]1))
3619, 19, 13, 13txunii 22824 . . . . . . . . . . . . . . . . 17 ((0[,]1) × (0[,]1)) = (II ×t II)
3736ntropn 22280 . . . . . . . . . . . . . . . 16 (((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) → ((int‘(II ×t II))‘𝑆) ∈ (II ×t II))
3821, 35, 37mp2an 689 . . . . . . . . . . . . . . 15 ((int‘(II ×t II))‘𝑆) ∈ (II ×t II)
3938a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → ((int‘(II ×t II))‘𝑆) ∈ (II ×t II))
402adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
413adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
424adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑃𝐵)
435adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → (𝐹𝑃) = (0𝐺0))
44 eqid 2736 . . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑏 ∈ (0[,]1))
46 simprl 768 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → 𝑎 ∈ (0[,]1))
471, 40, 41, 42, 43, 6, 7, 44, 45, 46cvmlift2lem10 33409 . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . . . . 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 22833 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ∈ Top ∧ II ∈ Top) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) → (𝑢 × 𝑣) ∈ (II ×t II))
5450, 50, 51, 52, 53syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑟𝑢𝑡𝑣) → 𝑡𝑣)
56 elunii 4854 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑡𝑣𝑣 ∈ II) → 𝑡 II)
5756, 13eleqtrrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑢 ∈ II)
61 simprl 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑏𝑢𝑎𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) ∧ (𝑟𝑢𝑡𝑣)) → 𝑣 ∈ II)
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 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → {𝑐} = {𝑤})
7372xpeq2d 5637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (𝑢 × {𝑐}) = (𝑢 × {𝑤}))
7473reseq2d 5910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (𝐾 ↾ (𝑢 × {𝑐})) = (𝐾 ↾ (𝑢 × {𝑤})))
7573oveq2d 7332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → ((II ×t II) ↾t (𝑢 × {𝑐})) = ((II ×t II) ↾t (𝑢 × {𝑤})))
7675oveq1d 7331 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → (((II ×t II) ↾t (𝑢 × {𝑐})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶))
7774, 76eleq12d 2831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 33410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 33410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 211 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . 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 1930 . . . . . . . . . . . . . . . . . . . . . . . 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 5613 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)}
9089, 33sseq12i 3960 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ {⟨𝑟, 𝑡⟩ ∣ (𝑟𝑢𝑡𝑣)} ⊆ {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))})
91 ssopab2bw 5479 . . . . . . . . . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑢 × 𝑣) ⊆ 𝑆 ↔ ∀𝑟𝑡((𝑟𝑢𝑡𝑣) → (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))))
9388, 92sylibr 233 . . . . . . . . . . . . . . . . . . . . . . 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 22289 . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . 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 5644 . . . . . . . . . . . . . . . . . . . . . . 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 3931 . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . 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 3201 . . . . . . . . . . . . . . . . . . 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 3444 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
105 opeq2 4815 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑎 → ⟨𝑏, 𝑤⟩ = ⟨𝑏, 𝑎⟩)
106105eleq1d 2821 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑎 → (⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆)))
107104, 106ralsn 4626 . . . . . . . . . . . . . . . . . 18 (∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑎⟩ ∈ ((int‘(II ×t II))‘𝑆))
108103, 107sylibr 233 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1))) → ∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
109108anassrs 468 . . . . . . . . . . . . . . . 16 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑏 ∈ (0[,]1)) → ∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
110109ralrimiva 3139 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (0[,]1)) → ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
111 dfss3 3918 . . . . . . . . . . . . . . . 16 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑢 ∈ ((0[,]1) × {𝑎})𝑢 ∈ ((int‘(II ×t II))‘𝑆))
112 eleq1 2824 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑏, 𝑤⟩ → (𝑢 ∈ ((int‘(II ×t II))‘𝑆) ↔ ⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆)))
113112ralxp 5770 . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 15 (((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆) ↔ ∀𝑏 ∈ (0[,]1)∀𝑤 ∈ {𝑎}⟨𝑏, 𝑤⟩ ∈ ((int‘(II ×t II))‘𝑆))
115110, 114sylibr 233 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → ((0[,]1) × {𝑎}) ⊆ ((int‘(II ×t II))‘𝑆))
116 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (0[,]1)) → 𝑎 ∈ (0[,]1))
11713, 13, 18, 20, 39, 115, 116txtube 22871 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (0[,]1)) → ∃𝑣 ∈ II (𝑎𝑣 ∧ ((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆)))
11836ntrss2 22288 . . . . . . . . . . . . . . . . . . 19 (((II ×t II) ∈ Top ∧ 𝑆 ⊆ ((0[,]1) × (0[,]1))) → ((int‘(II ×t II))‘𝑆) ⊆ 𝑆)
11921, 35, 118mp2an 689 . . . . . . . . . . . . . . . . . 18 ((int‘(II ×t II))‘𝑆) ⊆ 𝑆
120 sstr 3938 . . . . . . . . . . . . . . . . . 18 ((((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) ∧ ((int‘(II ×t II))‘𝑆) ⊆ 𝑆) → ((0[,]1) × 𝑣) ⊆ 𝑆)
121119, 120mpan2 688 . . . . . . . . . . . . . . . . 17 (((0[,]1) × 𝑣) ⊆ ((int‘(II ×t II))‘𝑆) → ((0[,]1) × 𝑣) ⊆ 𝑆)
122 df-xp 5613 . . . . . . . . . . . . . . . . . . 19 ((0[,]1) × 𝑣) = {⟨𝑟, 𝑡⟩ ∣ (𝑟 ∈ (0[,]1) ∧ 𝑡𝑣)}
123122, 33sseq12i 3960 . . . . . . . . . . . . . . . . . 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 5479 . . . . . . . . . . . . . . . . . . 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 3187 . . . . . . . . . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . . . . . 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 298 . . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . . 17 (((0[,]1) × 𝑣) ⊆ 𝑆 ↔ ∀𝑡𝑣𝑟 ∈ (0[,]1)(𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀)))
129121, 128sylib 217 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . 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 3082 . . . . . . . . . . . . . . . . . . 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 33399 . . . . . . . . . . . . . . . . . . . 20 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑎}) ⊆ 𝑀 → ((0[,]1) × {𝑡}) ⊆ 𝑀))
133 bicom 221 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
134133rexbii 3093 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀))
135134ralbii 3092 . . . . . . . . . . . . . . . . . . . . 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 33399 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑡}) ⊆ 𝑀 ↔ (𝑢 × {𝑎}) ⊆ 𝑀) → (((0[,]1) × {𝑡}) ⊆ 𝑀 → ((0[,]1) × {𝑎}) ⊆ 𝑀))
137135, 136sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (∀𝑟 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑡}) ⊆ 𝑀 → ((0[,]1) × {𝑎}) ⊆ 𝑀))
138132, 137impbid 211 . . . . . . . . . . . . . . . . . . 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}
141140rabeq2i 3424 . . . . . . . . . . . . . . . . . . . . 21 (𝑎𝐴 ↔ (𝑎 ∈ (0[,]1) ∧ ((0[,]1) × {𝑎}) ⊆ 𝑀))
142141baib 536 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (0[,]1) → (𝑎𝐴 ↔ ((0[,]1) × {𝑎}) ⊆ 𝑀))
143142ad3antlr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → (𝑎𝐴 ↔ ((0[,]1) × {𝑎}) ⊆ 𝑀))
144 elssuni 4882 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ II → 𝑣 II)
145144, 13sseqtrrdi 3981 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ II → 𝑣 ⊆ (0[,]1))
146145adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) → 𝑣 ⊆ (0[,]1))
147146sselda 3930 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑎 ∈ (0[,]1)) ∧ 𝑣 ∈ II) ∧ 𝑡𝑣) → 𝑡 ∈ (0[,]1))
148 sneq 4580 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑡 → {𝑎} = {𝑡})
149148xpeq2d 5637 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑡 → ((0[,]1) × {𝑎}) = ((0[,]1) × {𝑡}))
150149sseq1d 3961 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑡 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {𝑡}) ⊆ 𝑀))
151150, 140elrab2 3636 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐴 ↔ (𝑡 ∈ (0[,]1) ∧ ((0[,]1) × {𝑡}) ⊆ 𝑀))
152151baib 536 . . . . . . . . . . . . . . . . . . . 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, 154syl5ibr 245 . . . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . 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 3139 . . . . . . . . . . 11 (𝜑 → ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
162 ssrab2 4023 . . . . . . . . . . . . 13 {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ⊆ (0[,]1)
163140, 162eqsstri 3964 . . . . . . . . . . . 12 𝐴 ⊆ (0[,]1)
16413isclo 22318 . . . . . . . . . . . 12 ((II ∈ Top ∧ 𝐴 ⊆ (0[,]1)) → (𝐴 ∈ (II ∩ (Clsd‘II)) ↔ ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴))))
16519, 163, 164mp2an 689 . . . . . . . . . . 11 (𝐴 ∈ (II ∩ (Clsd‘II)) ↔ ∀𝑎 ∈ (0[,]1)∃𝑣 ∈ II (𝑎𝑣 ∧ ∀𝑡𝑣 (𝑎𝐴𝑡𝐴)))
166161, 165sylibr 233 . . . . . . . . . 10 (𝜑𝐴 ∈ (II ∩ (Clsd‘II)))
16716, 166sselid 3928 . . . . . . . . 9 (𝜑𝐴 ∈ II)
168 0elunit 13280 . . . . . . . . . . . 12 0 ∈ (0[,]1)
169168a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ∈ (0[,]1))
170 relxp 5625 . . . . . . . . . . . . 13 Rel ((0[,]1) × {0})
171170a1i 11 . . . . . . . . . . . 12 (𝜑 → Rel ((0[,]1) × {0}))
172 opelxp 5643 . . . . . . . . . . . . 13 (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) ↔ (𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}))
173 id 22 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ (0[,]1) → 𝑟 ∈ (0[,]1))
174 opelxpi 5644 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
1773adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝐺 ∈ ((II ×t II) Cn 𝐽))
1784adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → 𝑃𝐵)
1795adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ (0[,]1)) → (𝐹𝑃) = (0𝐺0))
180 simpr 485 . . . . . . . . . . . . . . . . . 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 33409 . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 0 ∈ 𝑣)
1858ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
186185ffnd 6638 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾 Fn ((0[,]1) × (0[,]1)))
187 fnov 7446 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐾 Fn ((0[,]1) × (0[,]1)) ↔ 𝐾 = (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)))
188186, 187sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝐾 = (𝑏 ∈ (0[,]1), 𝑤 ∈ (0[,]1) ↦ (𝑏𝐾𝑤)))
189188reseq1d 5909 . . . . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ∈ II)
191 elssuni 4882 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 ∈ II → 𝑢 II)
192191, 13sseqtrrdi 3981 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 ∈ II → 𝑢 ⊆ (0[,]1))
193190, 192syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝑢 ⊆ (0[,]1))
194169snssd 4753 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → {0} ⊆ (0[,]1))
195194ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → {0} ⊆ (0[,]1))
196 resmpo 7435 . . . . . . . . . . . . . . . . . . . . . . . . 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 3930 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) ∧ 𝑏𝑢) → 𝑏 ∈ (0[,]1))
199 simplll 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → 𝜑)
2001, 2, 3, 4, 5, 6, 7cvmlift2lem8 33407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 ∈ {0} → 𝑤 = 0)
204203oveq2d 7332 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {0} → (𝑏𝐾𝑤) = (𝑏𝐾0))
205204eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ {0} → ((𝑏𝐾𝑤) = (𝐻𝑏) ↔ (𝑏𝐾0) = (𝐻𝑏)))
206202, 205syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝑏𝐾𝑤)) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
209189, 197, 2083eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) = (𝑏𝑢, 𝑤 ∈ {0} ↦ (𝐻𝑏)))
210 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t 𝑢) = (II ↾t 𝑢)
211 iitopon 24122 . . . . . . . . . . . . . . . . . . . . . . . . . 26 II ∈ (TopOn‘(0[,]1))
212211a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → II ∈ (TopOn‘(0[,]1)))
213 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (II ↾t {0}) = (II ↾t {0})
214212, 212cnmpt1st 22899 . . . . . . . . . . . . . . . . . . . . . . . . . 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 33401 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22903 . . . . . . . . . . . . . . . . . . . . . . . . 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 22908 . . . . . . . . . . . . . . . . . . . . . . . 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 3444 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑢 ∈ V
221 snex 5368 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {0} ∈ V
222 txrest 22862 . . . . . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × {0})) = ((II ↾t 𝑢) ×t (II ↾t {0}))
224223oveq1i 7326 . . . . . . . . . . . . . . . . . . . . . . . 24 (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶) = (((II ↾t 𝑢) ×t (II ↾t {0})) Cn 𝐶)
225219, 224eleqtrrdi 2848 . . . . . . . . . . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝐾 ↾ (𝑢 × {0})) ∈ (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
227 sneq 4580 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = 0 → {𝑤} = {0})
228227xpeq2d 5637 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → (𝑢 × {𝑤}) = (𝑢 × {0}))
229228reseq2d 5910 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (𝐾 ↾ (𝑢 × {𝑤})) = (𝐾 ↾ (𝑢 × {0})))
230228oveq2d 7332 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 0 → ((II ×t II) ↾t (𝑢 × {𝑤})) = ((II ×t II) ↾t (𝑢 × {0})))
231230oveq1d 7331 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 0 → (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) = (((II ×t II) ↾t (𝑢 × {0})) Cn 𝐶))
232229, 231eleq12d 2831 . . . . . . . . . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . . . . . . . . . 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 5644 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑟𝑢 ∧ 0 ∈ 𝑣) → ⟨𝑟, 0⟩ ∈ (𝑢 × 𝑣))
236235adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ (𝑢 × 𝑣))
237 simplrr 775 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5622 . . . . . . . . . . . . . . . . . . . . . . . . . 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 22393 . . . . . . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((II ×t II) ↾t (𝑢 × 𝑣)))
244 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((II ×t II) ↾t (𝑢 × 𝑣)) = ((II ×t II) ↾t (𝑢 × 𝑣))
245244cncnpi 22509 . . . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → (𝑢 × 𝑣) ∈ (II ×t II))
251 isopn3i 22313 . . . . . . . . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑟 ∈ (0[,]1)) ∧ (𝑢 ∈ II ∧ 𝑣 ∈ II)) ∧ (𝑟𝑢 ∧ 0 ∈ 𝑣)) → ⟨𝑟, 0⟩ ∈ ((int‘(II ×t II))‘(𝑢 × 𝑣)))
25436, 1cnprest 22520 . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . . 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 3201 . . . . . . . . . . . . . . . . 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 6811 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑟, 0⟩ → (((II ×t II) CnP 𝐶)‘𝑧) = (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩))
263262eleq2d 2822 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑟, 0⟩ → (𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧) ↔ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘⟨𝑟, 0⟩)))
264263, 68elrab2 3636 . . . . . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {0} → 𝑎 = 0)
267266opeq2d 4821 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ = ⟨𝑟, 0⟩)
268267eleq1d 2821 . . . . . . . . . . . . . . 15 (𝑎 ∈ {0} → (⟨𝑟, 𝑎⟩ ∈ 𝑀 ↔ ⟨𝑟, 0⟩ ∈ 𝑀))
269265, 268syl5ibrcom 246 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (0[,]1)) → (𝑎 ∈ {0} → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
270269expimpd 454 . . . . . . . . . . . . 13 (𝜑 → ((𝑟 ∈ (0[,]1) ∧ 𝑎 ∈ {0}) → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
271172, 270biimtrid 241 . . . . . . . . . . . 12 (𝜑 → (⟨𝑟, 𝑎⟩ ∈ ((0[,]1) × {0}) → ⟨𝑟, 𝑎⟩ ∈ 𝑀))
272171, 271relssdv 5717 . . . . . . . . . . 11 (𝜑 → ((0[,]1) × {0}) ⊆ 𝑀)
273 sneq 4580 . . . . . . . . . . . . . 14 (𝑎 = 0 → {𝑎} = {0})
274273xpeq2d 5637 . . . . . . . . . . . . 13 (𝑎 = 0 → ((0[,]1) × {𝑎}) = ((0[,]1) × {0}))
275274sseq1d 3961 . . . . . . . . . . . 12 (𝑎 = 0 → (((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ((0[,]1) × {0}) ⊆ 𝑀))
276275, 140elrab2 3636 . . . . . . . . . . 11 (0 ∈ 𝐴 ↔ (0 ∈ (0[,]1) ∧ ((0[,]1) × {0}) ⊆ 𝑀))
277169, 272, 276sylanbrc 583 . . . . . . . . . 10 (𝜑 → 0 ∈ 𝐴)
278277ne0d 4279 . . . . . . . . 9 (𝜑𝐴 ≠ ∅)
279 inss2 4173 . . . . . . . . . 10 (II ∩ (Clsd‘II)) ⊆ (Clsd‘II)
280279, 166sselid 3928 . . . . . . . . 9 (𝜑𝐴 ∈ (Clsd‘II))
28113, 15, 167, 278, 280connclo 22646 . . . . . . . 8 (𝜑𝐴 = (0[,]1))
282281, 140eqtr3di 2791 . . . . . . 7 (𝜑 → (0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀})
283 rabid2 3431 . . . . . . 7 ((0[,]1) = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} ↔ ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
284282, 283sylib 217 . . . . . 6 (𝜑 → ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
285 iunss 4987 . . . . . 6 ( 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀 ↔ ∀𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
286284, 285sylibr 233 . . . . 5 (𝜑 𝑎 ∈ (0[,]1)((0[,]1) × {𝑎}) ⊆ 𝑀)
28712, 286eqsstrid 3978 . . . 4 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ 𝑀)
288287, 68sseqtrdi 3980 . . 3 (𝜑 → ((0[,]1) × (0[,]1)) ⊆ {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)})
289 ssrab 4016 . . . 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 497 . . 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 22822 . . . 4 ((II ∈ (TopOn‘(0[,]1)) ∧ II ∈ (TopOn‘(0[,]1))) → (II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1))))
293211, 211, 292mp2an 689 . . 3 (II ×t II) ∈ (TopOn‘((0[,]1) × (0[,]1)))
294 cvmtop1 33357 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2952, 294syl 17 . . . 4 (𝜑𝐶 ∈ Top)
2961toptopon 22146 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
297295, 296sylib 217 . . 3 (𝜑𝐶 ∈ (TopOn‘𝐵))
298 cncnp 22511 . . 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 710 1 (𝜑𝐾 ∈ ((II ×t II) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wal 1538   = wceq 1540  wcel 2105  wral 3061  wrex 3070  {crab 3403  Vcvv 3440  cdif 3893  cin 3895  wss 3896  c0 4266  𝒫 cpw 4544  {csn 4570  cop 4576   cuni 4849   ciun 4936  {copab 5148  cmpt 5169   × cxp 5605  ccnv 5606  cres 5609  cima 5610  ccom 5611  Rel wrel 5612   Fn wfn 6460  wf 6461  cfv 6465  crio 7272  (class class class)co 7316  cmpo 7318  0cc0 10950  1c1 10951  [,]cicc 13161  t crest 17205  Topctop 22122  TopOnctopon 22139  Clsdccld 22247  intcnt 22248  neicnei 22328   Cn ccn 22455   CnP ccnp 22456  Compccmp 22617  Conncconn 22642   ×t ctx 22791  Homeochmeo 22984  IIcii 24118   CovMap ccvm 33352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-inf2 9476  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027  ax-pre-sup 11028  ax-addf 11029  ax-mulf 11030
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-of 7574  df-om 7759  df-1st 7877  df-2nd 7878  df-supp 8026  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-2o 8346  df-er 8547  df-ec 8549  df-map 8666  df-ixp 8735  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-fsupp 9205  df-fi 9246  df-sup 9277  df-inf 9278  df-oi 9345  df-card 9774  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-div 11712  df-nn 12053  df-2 12115  df-3 12116  df-4 12117  df-5 12118  df-6 12119  df-7 12120  df-8 12121  df-9 12122  df-n0 12313  df-z 12399  df-dec 12517  df-uz 12662  df-q 12768  df-rp 12810  df-xneg 12927  df-xadd 12928  df-xmul 12929  df-ioo 13162  df-ico 13164  df-icc 13165  df-fz 13319  df-fzo 13462  df-fl 13591  df-seq 13801  df-exp 13862  df-hash 14124  df-cj 14886  df-re 14887  df-im 14888  df-sqrt 15022  df-abs 15023  df-clim 15273  df-sum 15474  df-struct 16922  df-sets 16939  df-slot 16957  df-ndx 16969  df-base 16987  df-ress 17016  df-plusg 17049  df-mulr 17050  df-starv 17051  df-sca 17052  df-vsca 17053  df-ip 17054  df-tset 17055  df-ple 17056  df-ds 17058  df-unif 17059  df-hom 17060  df-cco 17061  df-rest 17207  df-topn 17208  df-0g 17226  df-gsum 17227  df-topgen 17228  df-pt 17229  df-prds 17232  df-xrs 17287  df-qtop 17292  df-imas 17293  df-xps 17295  df-mre 17369  df-mrc 17370  df-acs 17372  df-mgm 18400  df-sgrp 18449  df-mnd 18460  df-submnd 18505  df-mulg 18774  df-cntz 18996  df-cmn 19460  df-psmet 20669  df-xmet 20670  df-met 20671  df-bl 20672  df-mopn 20673  df-cnfld 20678  df-top 22123  df-topon 22140  df-topsp 22162  df-bases 22176  df-cld 22250  df-ntr 22251  df-cls 22252  df-nei 22329  df-cn 22458  df-cnp 22459  df-cmp 22618  df-conn 22643  df-lly 22697  df-nlly 22698  df-tx 22793  df-hmeo 22986  df-xms 23553  df-ms 23554  df-tms 23555  df-ii 24120  df-htpy 24213  df-phtpy 24214  df-phtpc 24235  df-pconn 33318  df-sconn 33319  df-cvm 33353
This theorem is referenced by:  cvmlift2lem13  33412
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