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Theorem cvmlift3lem5 31066
Description: Lemma for cvmlift2 31059. (Contributed by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
cvmlift3.b 𝐵 = 𝐶
cvmlift3.y 𝑌 = 𝐾
cvmlift3.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k (𝜑𝐾 ∈ SConn)
cvmlift3.l (𝜑𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o (𝜑𝑂𝑌)
cvmlift3.g (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p (𝜑𝑃𝐵)
cvmlift3.e (𝜑 → (𝐹𝑃) = (𝐺𝑂))
cvmlift3.h 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
Assertion
Ref Expression
cvmlift3lem5 (𝜑 → (𝐹𝐻) = 𝐺)
Distinct variable groups:   𝑧,𝑓,𝑔,𝑥   𝑓,𝐽   𝑥,𝑔,𝐽   𝑓,𝐹,𝑔   𝑥,𝑧,𝐹   𝑓,𝐻,𝑔,𝑥,𝑧   𝐵,𝑓,𝑔,𝑥,𝑧   𝑓,𝐺,𝑔,𝑥,𝑧   𝐶,𝑓,𝑔,𝑥,𝑧   𝜑,𝑓,𝑥   𝑓,𝐾,𝑔,𝑥,𝑧   𝑃,𝑓,𝑔,𝑥,𝑧   𝑓,𝑂,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑔)   𝐽(𝑧)

Proof of Theorem cvmlift3lem5
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . 5 (𝐻𝑦) = (𝐻𝑦)
2 cvmlift3.b . . . . . 6 𝐵 = 𝐶
3 cvmlift3.y . . . . . 6 𝑌 = 𝐾
4 cvmlift3.f . . . . . 6 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
5 cvmlift3.k . . . . . 6 (𝜑𝐾 ∈ SConn)
6 cvmlift3.l . . . . . 6 (𝜑𝐾 ∈ 𝑛-Locally PConn)
7 cvmlift3.o . . . . . 6 (𝜑𝑂𝑌)
8 cvmlift3.g . . . . . 6 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
9 cvmlift3.p . . . . . 6 (𝜑𝑃𝐵)
10 cvmlift3.e . . . . . 6 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
11 cvmlift3.h . . . . . 6 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
122, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 31065 . . . . 5 ((𝜑𝑦𝑌) → ((𝐻𝑦) = (𝐻𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦))))
131, 12mpbii 223 . . . 4 ((𝜑𝑦𝑌) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦)))
14 df-3an 1038 . . . . . 6 (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦)) ↔ (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦) ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦)))
15 eqid 2621 . . . . . . . . . . . 12 (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))
164ad3antrrr 765 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17 simplr 791 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓 ∈ (II Cn 𝐾))
188ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19 cnco 21010 . . . . . . . . . . . . 13 ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑓) ∈ (II Cn 𝐽))
2017, 18, 19syl2anc 692 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺𝑓) ∈ (II Cn 𝐽))
219ad3antrrr 765 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑃𝐵)
22 simprl 793 . . . . . . . . . . . . . 14 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘0) = 𝑂)
2322fveq2d 6162 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘0)) = (𝐺𝑂))
24 iiuni 22624 . . . . . . . . . . . . . . . 16 (0[,]1) = II
2524, 3cnf 20990 . . . . . . . . . . . . . . 15 (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶𝑌)
2617, 25syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓:(0[,]1)⟶𝑌)
27 0elunit 12248 . . . . . . . . . . . . . 14 0 ∈ (0[,]1)
28 fvco3 6242 . . . . . . . . . . . . . 14 ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑓)‘0) = (𝐺‘(𝑓‘0)))
2926, 27, 28sylancl 693 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺𝑓)‘0) = (𝐺‘(𝑓‘0)))
3010ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹𝑃) = (𝐺𝑂))
3123, 29, 303eqtr4rd 2666 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹𝑃) = ((𝐺𝑓)‘0))
322, 15, 16, 20, 21, 31cvmliftiota 31044 . . . . . . . . . . 11 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) ∧ (𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺𝑓) ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘0) = 𝑃))
3332simp2d 1072 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺𝑓))
3433fveq1d 6160 . . . . . . . . 9 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))‘1) = ((𝐺𝑓)‘1))
3532simp1d 1071 . . . . . . . . . . 11 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶))
3624, 2cnf 20990 . . . . . . . . . . 11 ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
3735, 36syl 17 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
38 1elunit 12249 . . . . . . . . . 10 1 ∈ (0[,]1)
39 fvco3 6242 . . . . . . . . . 10 (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))‘1) = (𝐹‘((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1)))
4037, 38, 39sylancl 693 . . . . . . . . 9 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)))‘1) = (𝐹‘((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1)))
41 fvco3 6242 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶𝑌 ∧ 1 ∈ (0[,]1)) → ((𝐺𝑓)‘1) = (𝐺‘(𝑓‘1)))
4226, 38, 41sylancl 693 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺𝑓)‘1) = (𝐺‘(𝑓‘1)))
43 simprr 795 . . . . . . . . . . 11 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘1) = 𝑦)
4443fveq2d 6162 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘1)) = (𝐺𝑦))
4542, 44eqtrd 2655 . . . . . . . . 9 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺𝑓)‘1) = (𝐺𝑦))
4634, 40, 453eqtr3d 2663 . . . . . . . 8 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹‘((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1)) = (𝐺𝑦))
47 fveq2 6158 . . . . . . . . 9 (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦) → (𝐹‘((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1)) = (𝐹‘(𝐻𝑦)))
4847eqeq1d 2623 . . . . . . . 8 (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦) → ((𝐹‘((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1)) = (𝐺𝑦) ↔ (𝐹‘(𝐻𝑦)) = (𝐺𝑦)))
4946, 48syl5ibcom 235 . . . . . . 7 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦) → (𝐹‘(𝐻𝑦)) = (𝐺𝑦)))
5049expimpd 628 . . . . . 6 (((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → ((((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦) ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦)) → (𝐹‘(𝐻𝑦)) = (𝐺𝑦)))
5114, 50syl5bi 232 . . . . 5 (((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦)) → (𝐹‘(𝐻𝑦)) = (𝐺𝑦)))
5251rexlimdva 3026 . . . 4 ((𝜑𝑦𝑌) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑦)) → (𝐹‘(𝐻𝑦)) = (𝐺𝑦)))
5313, 52mpd 15 . . 3 ((𝜑𝑦𝑌) → (𝐹‘(𝐻𝑦)) = (𝐺𝑦))
5453mpteq2dva 4714 . 2 (𝜑 → (𝑦𝑌 ↦ (𝐹‘(𝐻𝑦))) = (𝑦𝑌 ↦ (𝐺𝑦)))
552, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem3 31064 . . . 4 (𝜑𝐻:𝑌𝐵)
5655ffvelrnda 6325 . . 3 ((𝜑𝑦𝑌) → (𝐻𝑦) ∈ 𝐵)
5755feqmptd 6216 . . 3 (𝜑𝐻 = (𝑦𝑌 ↦ (𝐻𝑦)))
58 cvmcn 31005 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
59 eqid 2621 . . . . . 6 𝐽 = 𝐽
602, 59cnf 20990 . . . . 5 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
614, 58, 603syl 18 . . . 4 (𝜑𝐹:𝐵 𝐽)
6261feqmptd 6216 . . 3 (𝜑𝐹 = (𝑤𝐵 ↦ (𝐹𝑤)))
63 fveq2 6158 . . 3 (𝑤 = (𝐻𝑦) → (𝐹𝑤) = (𝐹‘(𝐻𝑦)))
6456, 57, 62, 63fmptco 6362 . 2 (𝜑 → (𝐹𝐻) = (𝑦𝑌 ↦ (𝐹‘(𝐻𝑦))))
653, 59cnf 20990 . . . 4 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
668, 65syl 17 . . 3 (𝜑𝐺:𝑌 𝐽)
6766feqmptd 6216 . 2 (𝜑𝐺 = (𝑦𝑌 ↦ (𝐺𝑦)))
6854, 64, 673eqtr4d 2665 1 (𝜑 → (𝐹𝐻) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2909   cuni 4409  cmpt 4683  ccom 5088  wf 5853  cfv 5857  crio 6575  (class class class)co 6615  0cc0 9896  1c1 9897  [,]cicc 12136   Cn ccn 20968  𝑛-Locally cnlly 21208  IIcii 22618  PConncpconn 30962  SConncsconn 30963   CovMap ccvm 30998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975  ax-mulf 9976
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-ec 7704  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-starv 15896  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-unif 15905  df-hom 15906  df-cco 15907  df-rest 16023  df-topn 16024  df-0g 16042  df-gsum 16043  df-topgen 16044  df-pt 16045  df-prds 16048  df-xrs 16102  df-qtop 16107  df-imas 16108  df-xps 16110  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-mulg 17481  df-cntz 17690  df-cmn 18135  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-cnfld 19687  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cld 20763  df-ntr 20764  df-cls 20765  df-nei 20842  df-cn 20971  df-cnp 20972  df-cmp 21130  df-conn 21155  df-lly 21209  df-nlly 21210  df-tx 21305  df-hmeo 21498  df-xms 22065  df-ms 22066  df-tms 22067  df-ii 22620  df-htpy 22709  df-phtpy 22710  df-phtpc 22731  df-pco 22745  df-pconn 30964  df-sconn 30965  df-cvm 30999
This theorem is referenced by:  cvmlift3lem6  31067  cvmlift3lem7  31068  cvmlift3lem9  31070
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