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Theorem cvmlift3lem6 32859
Description: Lemma for cvmlift3 32863. (Contributed by Mario Carneiro, 9-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)))
cvmlift3lem7.s 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
cvmlift3lem7.1 (𝜑 → (𝐺𝑋) ∈ 𝐴)
cvmlift3lem7.2 (𝜑𝑇 ∈ (𝑆𝐴))
cvmlift3lem7.3 (𝜑𝑀 ⊆ (𝐺𝐴))
cvmlift3lem7.w 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
cvmlift3lem6.x (𝜑𝑋𝑀)
cvmlift3lem6.z (𝜑𝑍𝑀)
cvmlift3lem6.q (𝜑𝑄 ∈ (II Cn 𝐾))
cvmlift3lem6.r 𝑅 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑄) ∧ (𝑔‘0) = 𝑃))
cvmlift3lem6.1 (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))
cvmlift3lem6.n (𝜑𝑁 ∈ (II Cn (𝐾t 𝑀)))
cvmlift3lem6.2 (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
cvmlift3lem6.i 𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))
Assertion
Ref Expression
cvmlift3lem6 (𝜑 → (𝐻𝑍) ∈ 𝑊)
Distinct variable groups:   𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝐴   𝑓,𝑔,𝐼,𝑧   𝑔,𝑏,𝑥,𝐽,𝑐,𝑑,𝑓,𝑘,𝑠   𝐹,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠   𝑥,𝑧,𝐹   𝑓,𝑀,𝑔,𝑥   𝑓,𝑁,𝑔   𝐻,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑄,𝑓,𝑔   𝑆,𝑏,𝑓,𝑥   𝐵,𝑏,𝑑,𝑓,𝑔,𝑥,𝑧   𝑅,𝑔   𝑋,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝐺,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧   𝑇,𝑏,𝑐,𝑑,𝑠   𝑓,𝑍,𝑔,𝑥,𝑧   𝐶,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠,𝑥,𝑧   𝜑,𝑓,𝑥   𝐾,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧   𝑃,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑂,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧   𝑊,𝑐,𝑑,𝑓,𝑥
Allowed substitution hints:   𝜑(𝑧,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑)   𝐴(𝑥,𝑔)   𝐵(𝑘,𝑠,𝑐)   𝑃(𝑘,𝑠)   𝑄(𝑥,𝑧,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑅(𝑥,𝑧,𝑓,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝑇(𝑥,𝑧,𝑓,𝑔,𝑘)   𝐺(𝑠)   𝐻(𝑘,𝑠)   𝐼(𝑥,𝑘,𝑠,𝑏,𝑐,𝑑)   𝐽(𝑧)   𝐾(𝑘,𝑠,𝑑)   𝑀(𝑧,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑁(𝑥,𝑧,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑂(𝑘,𝑠,𝑑)   𝑊(𝑧,𝑔,𝑘,𝑠,𝑏)   𝑋(𝑘,𝑠)   𝑌(𝑘,𝑠,𝑏,𝑐,𝑑)   𝑍(𝑘,𝑠,𝑏,𝑐,𝑑)

Proof of Theorem cvmlift3lem6
StepHypRef Expression
1 cvmlift3lem6.q . . . . 5 (𝜑𝑄 ∈ (II Cn 𝐾))
2 cvmlift3.k . . . . . . . 8 (𝜑𝐾 ∈ SConn)
3 sconntop 32763 . . . . . . . 8 (𝐾 ∈ SConn → 𝐾 ∈ Top)
42, 3syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
5 cnrest2r 22040 . . . . . . 7 (𝐾 ∈ Top → (II Cn (𝐾t 𝑀)) ⊆ (II Cn 𝐾))
64, 5syl 17 . . . . . 6 (𝜑 → (II Cn (𝐾t 𝑀)) ⊆ (II Cn 𝐾))
7 cvmlift3lem6.n . . . . . 6 (𝜑𝑁 ∈ (II Cn (𝐾t 𝑀)))
86, 7sseldd 3878 . . . . 5 (𝜑𝑁 ∈ (II Cn 𝐾))
9 cvmlift3lem6.1 . . . . . . 7 (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))
109simp2d 1144 . . . . . 6 (𝜑 → (𝑄‘1) = 𝑋)
11 cvmlift3lem6.2 . . . . . . 7 (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
1211simpld 498 . . . . . 6 (𝜑 → (𝑁‘0) = 𝑋)
1310, 12eqtr4d 2776 . . . . 5 (𝜑 → (𝑄‘1) = (𝑁‘0))
141, 8, 13pcocn 23771 . . . 4 (𝜑 → (𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾))
151, 8pco0 23768 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = (𝑄‘0))
169simp1d 1143 . . . . 5 (𝜑 → (𝑄‘0) = 𝑂)
1715, 16eqtrd 2773 . . . 4 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂)
181, 8pco1 23769 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘1) = (𝑁‘1))
1911simprd 499 . . . . 5 (𝜑 → (𝑁‘1) = 𝑍)
2018, 19eqtrd 2773 . . . 4 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍)
21 cvmlift3.b . . . . . . . . . . 11 𝐵 = 𝐶
22 cvmlift3lem6.r . . . . . . . . . . 11 𝑅 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑄) ∧ (𝑔‘0) = 𝑃))
23 cvmlift3.f . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
24 cvmlift3.g . . . . . . . . . . . 12 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
25 cnco 22019 . . . . . . . . . . . 12 ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑄) ∈ (II Cn 𝐽))
261, 24, 25syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝐺𝑄) ∈ (II Cn 𝐽))
27 cvmlift3.p . . . . . . . . . . 11 (𝜑𝑃𝐵)
2816fveq2d 6680 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺𝑂))
29 iiuni 23635 . . . . . . . . . . . . . . 15 (0[,]1) = II
30 cvmlift3.y . . . . . . . . . . . . . . 15 𝑌 = 𝐾
3129, 30cnf 21999 . . . . . . . . . . . . . 14 (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
321, 31syl 17 . . . . . . . . . . . . 13 (𝜑𝑄:(0[,]1)⟶𝑌)
33 0elunit 12945 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
34 fvco3 6769 . . . . . . . . . . . . 13 ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
3532, 33, 34sylancl 589 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
36 cvmlift3.e . . . . . . . . . . . 12 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
3728, 35, 363eqtr4rd 2784 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = ((𝐺𝑄)‘0))
3821, 22, 23, 26, 27, 37cvmliftiota 32836 . . . . . . . . . 10 (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹𝑅) = (𝐺𝑄) ∧ (𝑅‘0) = 𝑃))
3938simp2d 1144 . . . . . . . . 9 (𝜑 → (𝐹𝑅) = (𝐺𝑄))
40 cvmlift3lem6.i . . . . . . . . . . 11 𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))
41 cnco 22019 . . . . . . . . . . . 12 ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑁) ∈ (II Cn 𝐽))
428, 24, 41syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝐺𝑁) ∈ (II Cn 𝐽))
43 cvmlift3.l . . . . . . . . . . . . 13 (𝜑𝐾 ∈ 𝑛-Locally PConn)
44 cvmlift3.o . . . . . . . . . . . . 13 (𝜑𝑂𝑌)
45 cvmlift3.h . . . . . . . . . . . . 13 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
4621, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem3 32856 . . . . . . . . . . . 12 (𝜑𝐻:𝑌𝐵)
47 cvmlift3lem7.3 . . . . . . . . . . . . . 14 (𝜑𝑀 ⊆ (𝐺𝐴))
48 cnvimass 5923 . . . . . . . . . . . . . . 15 (𝐺𝐴) ⊆ dom 𝐺
49 eqid 2738 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5030, 49cnf 21999 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
5124, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑌 𝐽)
5248, 51fssdm 6524 . . . . . . . . . . . . . 14 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
5347, 52sstrd 3887 . . . . . . . . . . . . 13 (𝜑𝑀𝑌)
54 cvmlift3lem6.x . . . . . . . . . . . . 13 (𝜑𝑋𝑀)
5553, 54sseldd 3878 . . . . . . . . . . . 12 (𝜑𝑋𝑌)
5646, 55ffvelrnd 6864 . . . . . . . . . . 11 (𝜑 → (𝐻𝑋) ∈ 𝐵)
5712fveq2d 6680 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺𝑋))
5829, 30cnf 21999 . . . . . . . . . . . . . 14 (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
598, 58syl 17 . . . . . . . . . . . . 13 (𝜑𝑁:(0[,]1)⟶𝑌)
60 fvco3 6769 . . . . . . . . . . . . 13 ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
6159, 33, 60sylancl 589 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
62 fvco3 6769 . . . . . . . . . . . . . 14 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6346, 55, 62syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6421, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem5 32858 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐻) = 𝐺)
6564fveq1d 6678 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
6663, 65eqtr3d 2775 . . . . . . . . . . . 12 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
6757, 61, 663eqtr4rd 2784 . . . . . . . . . . 11 (𝜑 → (𝐹‘(𝐻𝑋)) = ((𝐺𝑁)‘0))
6821, 40, 23, 42, 56, 67cvmliftiota 32836 . . . . . . . . . 10 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹𝐼) = (𝐺𝑁) ∧ (𝐼‘0) = (𝐻𝑋)))
6968simp2d 1144 . . . . . . . . 9 (𝜑 → (𝐹𝐼) = (𝐺𝑁))
7039, 69oveq12d 7190 . . . . . . . 8 (𝜑 → ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
7138simp1d 1143 . . . . . . . . 9 (𝜑𝑅 ∈ (II Cn 𝐶))
7268simp1d 1143 . . . . . . . . 9 (𝜑𝐼 ∈ (II Cn 𝐶))
739simp3d 1145 . . . . . . . . . 10 (𝜑 → (𝑅‘1) = (𝐻𝑋))
7468simp3d 1145 . . . . . . . . . 10 (𝜑 → (𝐼‘0) = (𝐻𝑋))
7573, 74eqtr4d 2776 . . . . . . . . 9 (𝜑 → (𝑅‘1) = (𝐼‘0))
76 cvmcn 32797 . . . . . . . . . 10 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
7723, 76syl 17 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
7871, 72, 75, 77copco 23772 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)))
791, 8, 13, 24copco 23772 . . . . . . . 8 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
8070, 78, 793eqtr4d 2783 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
8171, 72pco0 23768 . . . . . . . 8 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = (𝑅‘0))
8238simp3d 1145 . . . . . . . 8 (𝜑 → (𝑅‘0) = 𝑃)
8381, 82eqtrd 2773 . . . . . . 7 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃)
8471, 72, 75pcocn 23771 . . . . . . . 8 (𝜑 → (𝑅(*𝑝𝐶)𝐼) ∈ (II Cn 𝐶))
85 cnco 22019 . . . . . . . . . 10 (((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8614, 24, 85syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8717fveq2d 6680 . . . . . . . . . 10 (𝜑 → (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)) = (𝐺𝑂))
8829, 30cnf 21999 . . . . . . . . . . . 12 ((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) → (𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌)
8914, 88syl 17 . . . . . . . . . . 11 (𝜑 → (𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌)
90 fvco3 6769 . . . . . . . . . . 11 (((𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9189, 33, 90sylancl 589 . . . . . . . . . 10 (𝜑 → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9287, 91, 363eqtr4rd 2784 . . . . . . . . 9 (𝜑 → (𝐹𝑃) = ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0))
9321cvmlift 32834 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽)) ∧ (𝑃𝐵 ∧ (𝐹𝑃) = ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0))) → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
9423, 86, 27, 92, 93syl22anc 838 . . . . . . . 8 (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
95 coeq2 5701 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝐹𝑔) = (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)))
9695eqeq1d 2740 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ↔ (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
97 fveq1 6675 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝑔‘0) = ((𝑅(*𝑝𝐶)𝐼)‘0))
9897eqeq1d 2740 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃))
9996, 98anbi12d 634 . . . . . . . . 9 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃)))
10099riota2 7155 . . . . . . . 8 (((𝑅(*𝑝𝐶)𝐼) ∈ (II Cn 𝐶) ∧ ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) → (((𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃) ↔ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*𝑝𝐶)𝐼)))
10184, 94, 100syl2anc 587 . . . . . . 7 (𝜑 → (((𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃) ↔ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*𝑝𝐶)𝐼)))
10280, 83, 101mpbi2and 712 . . . . . 6 (𝜑 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*𝑝𝐶)𝐼))
103102fveq1d 6678 . . . . 5 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑅(*𝑝𝐶)𝐼)‘1))
10471, 72pco1 23769 . . . . 5 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘1) = (𝐼‘1))
105103, 104eqtrd 2773 . . . 4 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106 fveq1 6675 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘0) = ((𝑄(*𝑝𝐾)𝑁)‘0))
107106eqeq1d 2740 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂))
108 fveq1 6675 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘1) = ((𝑄(*𝑝𝐾)𝑁)‘1))
109108eqeq1d 2740 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍))
110 coeq2 5701 . . . . . . . . . . 11 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝐺𝑓) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
111110eqeq2d 2749 . . . . . . . . . 10 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
112111anbi1d 633 . . . . . . . . 9 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
113112riotabidv 7131 . . . . . . . 8 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
114113fveq1d 6678 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1))
115114eqeq1d 2740 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1) ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)))
116107, 109, 1153anbi123d 1437 . . . . 5 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)) ↔ (((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂 ∧ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))))
117116rspcev 3526 . . . 4 (((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) ∧ (((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂 ∧ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)))
11814, 17, 20, 105, 117syl13anc 1373 . . 3 (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)))
119 cvmlift3lem6.z . . . . 5 (𝜑𝑍𝑀)
12053, 119sseldd 3878 . . . 4 (𝜑𝑍𝑌)
12121, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem4 32857 . . . 4 ((𝜑𝑍𝑌) → ((𝐻𝑍) = (𝐼‘1) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))))
122120, 121mpdan 687 . . 3 (𝜑 → ((𝐻𝑍) = (𝐼‘1) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))))
123118, 122mpbird 260 . 2 (𝜑 → (𝐻𝑍) = (𝐼‘1))
124 iiconn 23641 . . . . 5 II ∈ Conn
125124a1i 11 . . . 4 (𝜑 → II ∈ Conn)
126 cvmtop1 32795 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
12723, 126syl 17 . . . . . . 7 (𝜑𝐶 ∈ Top)
12821toptopon 21670 . . . . . . 7 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129127, 128sylib 221 . . . . . 6 (𝜑𝐶 ∈ (TopOn‘𝐵))
13069rneqd 5781 . . . . . . . . 9 (𝜑 → ran (𝐹𝐼) = ran (𝐺𝑁))
131 rnco2 6086 . . . . . . . . 9 ran (𝐹𝐼) = (𝐹 “ ran 𝐼)
132 rnco2 6086 . . . . . . . . 9 ran (𝐺𝑁) = (𝐺 “ ran 𝑁)
133130, 131, 1323eqtr3g 2796 . . . . . . . 8 (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134 iitopon 23633 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
135134a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
13630toptopon 21670 . . . . . . . . . . . . . 14 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
1374, 136sylib 221 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ (TopOn‘𝑌))
138 resttopon 21914 . . . . . . . . . . . . 13 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
139137, 53, 138syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
140 cnf2 22002 . . . . . . . . . . . 12 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ 𝑁 ∈ (II Cn (𝐾t 𝑀))) → 𝑁:(0[,]1)⟶𝑀)
141135, 139, 7, 140syl3anc 1372 . . . . . . . . . . 11 (𝜑𝑁:(0[,]1)⟶𝑀)
142141frnd 6512 . . . . . . . . . 10 (𝜑 → ran 𝑁𝑀)
143142, 47sstrd 3887 . . . . . . . . 9 (𝜑 → ran 𝑁 ⊆ (𝐺𝐴))
14451ffund 6508 . . . . . . . . . 10 (𝜑 → Fun 𝐺)
145143, 48sstrdi 3889 . . . . . . . . . 10 (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146 funimass3 6833 . . . . . . . . . 10 ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
147144, 145, 146syl2anc 587 . . . . . . . . 9 (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
148143, 147mpbird 260 . . . . . . . 8 (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149133, 148eqsstrd 3915 . . . . . . 7 (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
15021, 49cnf 21999 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
15177, 150syl 17 . . . . . . . . 9 (𝜑𝐹:𝐵 𝐽)
152151ffund 6508 . . . . . . . 8 (𝜑 → Fun 𝐹)
15329, 21cnf 21999 . . . . . . . . . . 11 (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
15472, 153syl 17 . . . . . . . . . 10 (𝜑𝐼:(0[,]1)⟶𝐵)
155154frnd 6512 . . . . . . . . 9 (𝜑 → ran 𝐼𝐵)
156151fdmd 6515 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐵)
157155, 156sseqtrrd 3918 . . . . . . . 8 (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158 funimass3 6833 . . . . . . . 8 ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
159152, 157, 158syl2anc 587 . . . . . . 7 (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
160149, 159mpbid 235 . . . . . 6 (𝜑 → ran 𝐼 ⊆ (𝐹𝐴))
161 cnvimass 5923 . . . . . . 7 (𝐹𝐴) ⊆ dom 𝐹
162161, 151fssdm 6524 . . . . . 6 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
163 cnrest2 22039 . . . . . 6 ((𝐶 ∈ (TopOn‘𝐵) ∧ ran 𝐼 ⊆ (𝐹𝐴) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
164129, 160, 162, 163syl3anc 1372 . . . . 5 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
16572, 164mpbid 235 . . . 4 (𝜑𝐼 ∈ (II Cn (𝐶t (𝐹𝐴))))
166 cvmlift3lem7.2 . . . . . . 7 (𝜑𝑇 ∈ (𝑆𝐴))
167 cvmlift3lem7.s . . . . . . . 8 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
168167cvmsss 32802 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
169166, 168syl 17 . . . . . 6 (𝜑𝑇𝐶)
170 cvmlift3lem7.1 . . . . . . . . 9 (𝜑 → (𝐺𝑋) ∈ 𝐴)
17166, 170eqeltrd 2833 . . . . . . . 8 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
172 cvmlift3lem7.w . . . . . . . . 9 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
173167, 21, 172cvmsiota 32812 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
17423, 166, 56, 171, 173syl13anc 1373 . . . . . . 7 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
175174simpld 498 . . . . . 6 (𝜑𝑊𝑇)
176169, 175sseldd 3878 . . . . 5 (𝜑𝑊𝐶)
177 elssuni 4828 . . . . . . 7 (𝑊𝑇𝑊 𝑇)
178175, 177syl 17 . . . . . 6 (𝜑𝑊 𝑇)
179167cvmsuni 32804 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇 = (𝐹𝐴))
180166, 179syl 17 . . . . . 6 (𝜑 𝑇 = (𝐹𝐴))
181178, 180sseqtrd 3917 . . . . 5 (𝜑𝑊 ⊆ (𝐹𝐴))
182167cvmsrcl 32799 . . . . . . . 8 (𝑇 ∈ (𝑆𝐴) → 𝐴𝐽)
183166, 182syl 17 . . . . . . 7 (𝜑𝐴𝐽)
184 cnima 22018 . . . . . . 7 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐶)
18577, 183, 184syl2anc 587 . . . . . 6 (𝜑 → (𝐹𝐴) ∈ 𝐶)
186 restopn2 21930 . . . . . 6 ((𝐶 ∈ Top ∧ (𝐹𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
187127, 185, 186syl2anc 587 . . . . 5 (𝜑 → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
188176, 181, 187mpbir2and 713 . . . 4 (𝜑𝑊 ∈ (𝐶t (𝐹𝐴)))
189167cvmscld 32808 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → 𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19023, 166, 175, 189syl3anc 1372 . . . 4 (𝜑𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19133a1i 11 . . . 4 (𝜑 → 0 ∈ (0[,]1))
192174simprd 499 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝑊)
19374, 192eqeltrd 2833 . . . 4 (𝜑 → (𝐼‘0) ∈ 𝑊)
19429, 125, 165, 188, 190, 191, 193conncn 22179 . . 3 (𝜑𝐼:(0[,]1)⟶𝑊)
195 1elunit 12946 . . 3 1 ∈ (0[,]1)
196 ffvelrn 6861 . . 3 ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197194, 195, 196sylancl 589 . 2 (𝜑 → (𝐼‘1) ∈ 𝑊)
198123, 197eqeltrd 2833 1 (𝜑 → (𝐻𝑍) ∈ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wrex 3054  ∃!wreu 3055  {crab 3057  cdif 3840  cin 3842  wss 3843  c0 4211  𝒫 cpw 4488  {csn 4516   cuni 4796  cmpt 5110  ccnv 5524  dom cdm 5525  ran crn 5526  cres 5527  cima 5528  ccom 5529  Fun wfun 6333  wf 6335  cfv 6339  crio 7128  (class class class)co 7172  0cc0 10617  1c1 10618  [,]cicc 12826  t crest 16799  Topctop 21646  TopOnctopon 21663  Clsdccld 21769   Cn ccn 21977  Conncconn 22164  𝑛-Locally cnlly 22218  Homeochmeo 22506  IIcii 23629  *𝑝cpco 23754  PConncpconn 32754  SConncsconn 32755   CovMap ccvm 32790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-inf2 9179  ax-cnex 10673  ax-resscn 10674  ax-1cn 10675  ax-icn 10676  ax-addcl 10677  ax-addrcl 10678  ax-mulcl 10679  ax-mulrcl 10680  ax-mulcom 10681  ax-addass 10682  ax-mulass 10683  ax-distr 10684  ax-i2m1 10685  ax-1ne0 10686  ax-1rid 10687  ax-rnegex 10688  ax-rrecex 10689  ax-cnre 10690  ax-pre-lttri 10691  ax-pre-lttrn 10692  ax-pre-ltadd 10693  ax-pre-mulgt0 10694  ax-pre-sup 10695  ax-addf 10696  ax-mulf 10697
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-of 7427  df-om 7602  df-1st 7716  df-2nd 7717  df-supp 7859  df-wrecs 7978  df-recs 8039  df-rdg 8077  df-1o 8133  df-2o 8134  df-er 8322  df-ec 8324  df-map 8441  df-ixp 8510  df-en 8558  df-dom 8559  df-sdom 8560  df-fin 8561  df-fsupp 8909  df-fi 8950  df-sup 8981  df-inf 8982  df-oi 9049  df-card 9443  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-sub 10952  df-neg 10953  df-div 11378  df-nn 11719  df-2 11781  df-3 11782  df-4 11783  df-5 11784  df-6 11785  df-7 11786  df-8 11787  df-9 11788  df-n0 11979  df-z 12065  df-dec 12182  df-uz 12327  df-q 12433  df-rp 12475  df-xneg 12592  df-xadd 12593  df-xmul 12594  df-ioo 12827  df-ico 12829  df-icc 12830  df-fz 12984  df-fzo 13127  df-fl 13255  df-seq 13463  df-exp 13524  df-hash 13785  df-cj 14550  df-re 14551  df-im 14552  df-sqrt 14686  df-abs 14687  df-clim 14937  df-sum 15138  df-struct 16590  df-ndx 16591  df-slot 16592  df-base 16594  df-sets 16595  df-ress 16596  df-plusg 16683  df-mulr 16684  df-starv 16685  df-sca 16686  df-vsca 16687  df-ip 16688  df-tset 16689  df-ple 16690  df-ds 16692  df-unif 16693  df-hom 16694  df-cco 16695  df-rest 16801  df-topn 16802  df-0g 16820  df-gsum 16821  df-topgen 16822  df-pt 16823  df-prds 16826  df-xrs 16880  df-qtop 16885  df-imas 16886  df-xps 16888  df-mre 16962  df-mrc 16963  df-acs 16965  df-mgm 17970  df-sgrp 18019  df-mnd 18030  df-submnd 18075  df-mulg 18345  df-cntz 18567  df-cmn 19028  df-psmet 20211  df-xmet 20212  df-met 20213  df-bl 20214  df-mopn 20215  df-cnfld 20220  df-top 21647  df-topon 21664  df-topsp 21686  df-bases 21699  df-cld 21772  df-ntr 21773  df-cls 21774  df-nei 21851  df-cn 21980  df-cnp 21981  df-cmp 22140  df-conn 22165  df-lly 22219  df-nlly 22220  df-tx 22315  df-hmeo 22508  df-xms 23075  df-ms 23076  df-tms 23077  df-ii 23631  df-htpy 23724  df-phtpy 23725  df-phtpc 23746  df-pco 23759  df-pconn 32756  df-sconn 32757  df-cvm 32791
This theorem is referenced by:  cvmlift3lem7  32860
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