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Theorem cvmlift3lem5 32655
Description: Lemma for cvmlift2 32648. (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 2824 . . . . 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 32654 . . . . 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 236 . . . 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 1086 . . . . . 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 2824 . . . . . . . . . . . 12 (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))
164ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17 simplr 768 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓 ∈ (II Cn 𝐾))
188ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19 cnco 21880 . . . . . . . . . . . . 13 ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑓) ∈ (II Cn 𝐽))
2017, 18, 19syl2anc 587 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺𝑓) ∈ (II Cn 𝐽))
219ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑃𝐵)
22 simprl 770 . . . . . . . . . . . . . 14 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘0) = 𝑂)
2322fveq2d 6667 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘0)) = (𝐺𝑂))
24 iiuni 23495 . . . . . . . . . . . . . . . 16 (0[,]1) = II
2524, 3cnf 21860 . . . . . . . . . . . . . . 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 12858 . . . . . . . . . . . . . 14 0 ∈ (0[,]1)
28 fvco3 6753 . . . . . . . . . . . . . 14 ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑓)‘0) = (𝐺‘(𝑓‘0)))
2926, 27, 28sylancl 589 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺𝑓)‘0) = (𝐺‘(𝑓‘0)))
3010ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹𝑃) = (𝐺𝑂))
3123, 29, 303eqtr4rd 2870 . . . . . . . . . . . 12 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹𝑃) = ((𝐺𝑓)‘0))
322, 15, 16, 20, 21, 31cvmliftiota 32633 . . . . . . . . . . 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 1140 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺𝑓))
3433fveq1d 6665 . . . . . . . . 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 1139 . . . . . . . . . . 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 21860 . . . . . . . . . . 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 12859 . . . . . . . . . 10 1 ∈ (0[,]1)
39 fvco3 6753 . . . . . . . . . 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 589 . . . . . . . . 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 6753 . . . . . . . . . . 11 ((𝑓:(0[,]1)⟶𝑌 ∧ 1 ∈ (0[,]1)) → ((𝐺𝑓)‘1) = (𝐺‘(𝑓‘1)))
4226, 38, 41sylancl 589 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺𝑓)‘1) = (𝐺‘(𝑓‘1)))
43 simprr 772 . . . . . . . . . . 11 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘1) = 𝑦)
4443fveq2d 6667 . . . . . . . . . 10 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘1)) = (𝐺𝑦))
4542, 44eqtrd 2859 . . . . . . . . 9 ((((𝜑𝑦𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺𝑓)‘1) = (𝐺𝑦))
4634, 40, 453eqtr3d 2867 . . . . . . . 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 fveqeq2 6672 . . . . . . . 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)))
4846, 47syl5ibcom 248 . . . . . . 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)))
4948expimpd 457 . . . . . 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)))
5014, 49syl5bi 245 . . . . 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)))
5150rexlimdva 3276 . . . 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)))
5213, 51mpd 15 . . 3 ((𝜑𝑦𝑌) → (𝐹‘(𝐻𝑦)) = (𝐺𝑦))
5352mpteq2dva 5148 . 2 (𝜑 → (𝑦𝑌 ↦ (𝐹‘(𝐻𝑦))) = (𝑦𝑌 ↦ (𝐺𝑦)))
542, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem3 32653 . . . 4 (𝜑𝐻:𝑌𝐵)
5554ffvelrnda 6844 . . 3 ((𝜑𝑦𝑌) → (𝐻𝑦) ∈ 𝐵)
5654feqmptd 6726 . . 3 (𝜑𝐻 = (𝑦𝑌 ↦ (𝐻𝑦)))
57 cvmcn 32594 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
58 eqid 2824 . . . . . 6 𝐽 = 𝐽
592, 58cnf 21860 . . . . 5 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
604, 57, 593syl 18 . . . 4 (𝜑𝐹:𝐵 𝐽)
6160feqmptd 6726 . . 3 (𝜑𝐹 = (𝑤𝐵 ↦ (𝐹𝑤)))
62 fveq2 6663 . . 3 (𝑤 = (𝐻𝑦) → (𝐹𝑤) = (𝐹‘(𝐻𝑦)))
6355, 56, 61, 62fmptco 6884 . 2 (𝜑 → (𝐹𝐻) = (𝑦𝑌 ↦ (𝐹‘(𝐻𝑦))))
643, 58cnf 21860 . . . 4 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
658, 64syl 17 . . 3 (𝜑𝐺:𝑌 𝐽)
6665feqmptd 6726 . 2 (𝜑𝐺 = (𝑦𝑌 ↦ (𝐺𝑦)))
6753, 63, 663eqtr4d 2869 1 (𝜑 → (𝐹𝐻) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134   cuni 4824  cmpt 5133  ccom 5547  wf 6341  cfv 6345  crio 7108  (class class class)co 7151  0cc0 10537  1c1 10538  [,]cicc 12740   Cn ccn 21838  𝑛-Locally cnlly 22079  IIcii 23489  PConncpconn 32551  SConncsconn 32552   CovMap ccvm 32587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-inf2 9103  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-isom 6354  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7405  df-om 7577  df-1st 7686  df-2nd 7687  df-supp 7829  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-2o 8101  df-oadd 8104  df-er 8287  df-ec 8289  df-map 8406  df-ixp 8460  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-fsupp 8833  df-fi 8874  df-sup 8905  df-inf 8906  df-oi 8973  df-card 9367  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11637  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12897  df-fzo 13040  df-fl 13168  df-seq 13376  df-exp 13437  df-hash 13698  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-rest 16698  df-topn 16699  df-0g 16717  df-gsum 16718  df-topgen 16719  df-pt 16720  df-prds 16723  df-xrs 16777  df-qtop 16782  df-imas 16783  df-xps 16785  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-submnd 17959  df-mulg 18227  df-cntz 18449  df-cmn 18910  df-psmet 20092  df-xmet 20093  df-met 20094  df-bl 20095  df-mopn 20096  df-cnfld 20101  df-top 21508  df-topon 21525  df-topsp 21547  df-bases 21560  df-cld 21633  df-ntr 21634  df-cls 21635  df-nei 21712  df-cn 21841  df-cnp 21842  df-cmp 22001  df-conn 22026  df-lly 22080  df-nlly 22081  df-tx 22176  df-hmeo 22369  df-xms 22936  df-ms 22937  df-tms 22938  df-ii 23491  df-htpy 23584  df-phtpy 23585  df-phtpc 23606  df-pco 23619  df-pconn 32553  df-sconn 32554  df-cvm 32588
This theorem is referenced by:  cvmlift3lem6  32656  cvmlift3lem7  32657  cvmlift3lem9  32659
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