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Theorem cvmlift3lem6 34967
Description: Lemma for cvmlift3 34971. (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 34871 . . . . . . . 8 (𝐾 ∈ SConn → 𝐾 ∈ Top)
42, 3syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
5 cnrest2r 23211 . . . . . . 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 3983 . . . . 5 (𝜑𝑁 ∈ (II Cn 𝐾))
9 cvmlift3lem6.1 . . . . . . 7 (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))
109simp2d 1140 . . . . . 6 (𝜑 → (𝑄‘1) = 𝑋)
11 cvmlift3lem6.2 . . . . . . 7 (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
1211simpld 493 . . . . . 6 (𝜑 → (𝑁‘0) = 𝑋)
1310, 12eqtr4d 2771 . . . . 5 (𝜑 → (𝑄‘1) = (𝑁‘0))
141, 8, 13pcocn 24964 . . . 4 (𝜑 → (𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾))
151, 8pco0 24961 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = (𝑄‘0))
169simp1d 1139 . . . . 5 (𝜑 → (𝑄‘0) = 𝑂)
1715, 16eqtrd 2768 . . . 4 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂)
181, 8pco1 24962 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘1) = (𝑁‘1))
1911simprd 494 . . . . 5 (𝜑 → (𝑁‘1) = 𝑍)
2018, 19eqtrd 2768 . . . 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 23190 . . . . . . . . . . . 12 ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑄) ∈ (II Cn 𝐽))
261, 24, 25syl2anc 582 . . . . . . . . . . 11 (𝜑 → (𝐺𝑄) ∈ (II Cn 𝐽))
27 cvmlift3.p . . . . . . . . . . 11 (𝜑𝑃𝐵)
2816fveq2d 6906 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺𝑂))
29 iiuni 24821 . . . . . . . . . . . . . . 15 (0[,]1) = II
30 cvmlift3.y . . . . . . . . . . . . . . 15 𝑌 = 𝐾
3129, 30cnf 23170 . . . . . . . . . . . . . 14 (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
321, 31syl 17 . . . . . . . . . . . . 13 (𝜑𝑄:(0[,]1)⟶𝑌)
33 0elunit 13486 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
34 fvco3 7002 . . . . . . . . . . . . 13 ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
3532, 33, 34sylancl 584 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
36 cvmlift3.e . . . . . . . . . . . 12 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
3728, 35, 363eqtr4rd 2779 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = ((𝐺𝑄)‘0))
3821, 22, 23, 26, 27, 37cvmliftiota 34944 . . . . . . . . . 10 (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹𝑅) = (𝐺𝑄) ∧ (𝑅‘0) = 𝑃))
3938simp2d 1140 . . . . . . . . 9 (𝜑 → (𝐹𝑅) = (𝐺𝑄))
40 cvmlift3lem6.i . . . . . . . . . . 11 𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))
41 cnco 23190 . . . . . . . . . . . 12 ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑁) ∈ (II Cn 𝐽))
428, 24, 41syl2anc 582 . . . . . . . . . . 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 34964 . . . . . . . . . . . 12 (𝜑𝐻:𝑌𝐵)
47 cvmlift3lem7.3 . . . . . . . . . . . . . 14 (𝜑𝑀 ⊆ (𝐺𝐴))
48 cnvimass 6090 . . . . . . . . . . . . . . 15 (𝐺𝐴) ⊆ dom 𝐺
49 eqid 2728 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5030, 49cnf 23170 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
5124, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑌 𝐽)
5248, 51fssdm 6747 . . . . . . . . . . . . . 14 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
5347, 52sstrd 3992 . . . . . . . . . . . . 13 (𝜑𝑀𝑌)
54 cvmlift3lem6.x . . . . . . . . . . . . 13 (𝜑𝑋𝑀)
5553, 54sseldd 3983 . . . . . . . . . . . 12 (𝜑𝑋𝑌)
5646, 55ffvelcdmd 7100 . . . . . . . . . . 11 (𝜑 → (𝐻𝑋) ∈ 𝐵)
5712fveq2d 6906 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺𝑋))
5829, 30cnf 23170 . . . . . . . . . . . . . 14 (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
598, 58syl 17 . . . . . . . . . . . . 13 (𝜑𝑁:(0[,]1)⟶𝑌)
60 fvco3 7002 . . . . . . . . . . . . 13 ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
6159, 33, 60sylancl 584 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
62 fvco3 7002 . . . . . . . . . . . . . 14 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6346, 55, 62syl2anc 582 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6421, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem5 34966 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐻) = 𝐺)
6564fveq1d 6904 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
6663, 65eqtr3d 2770 . . . . . . . . . . . 12 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
6757, 61, 663eqtr4rd 2779 . . . . . . . . . . 11 (𝜑 → (𝐹‘(𝐻𝑋)) = ((𝐺𝑁)‘0))
6821, 40, 23, 42, 56, 67cvmliftiota 34944 . . . . . . . . . 10 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹𝐼) = (𝐺𝑁) ∧ (𝐼‘0) = (𝐻𝑋)))
6968simp2d 1140 . . . . . . . . 9 (𝜑 → (𝐹𝐼) = (𝐺𝑁))
7039, 69oveq12d 7444 . . . . . . . 8 (𝜑 → ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
7138simp1d 1139 . . . . . . . . 9 (𝜑𝑅 ∈ (II Cn 𝐶))
7268simp1d 1139 . . . . . . . . 9 (𝜑𝐼 ∈ (II Cn 𝐶))
739simp3d 1141 . . . . . . . . . 10 (𝜑 → (𝑅‘1) = (𝐻𝑋))
7468simp3d 1141 . . . . . . . . . 10 (𝜑 → (𝐼‘0) = (𝐻𝑋))
7573, 74eqtr4d 2771 . . . . . . . . 9 (𝜑 → (𝑅‘1) = (𝐼‘0))
76 cvmcn 34905 . . . . . . . . . 10 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
7723, 76syl 17 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
7871, 72, 75, 77copco 24965 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)))
791, 8, 13, 24copco 24965 . . . . . . . 8 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
8070, 78, 793eqtr4d 2778 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
8171, 72pco0 24961 . . . . . . . 8 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = (𝑅‘0))
8238simp3d 1141 . . . . . . . 8 (𝜑 → (𝑅‘0) = 𝑃)
8381, 82eqtrd 2768 . . . . . . 7 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃)
8471, 72, 75pcocn 24964 . . . . . . . 8 (𝜑 → (𝑅(*𝑝𝐶)𝐼) ∈ (II Cn 𝐶))
85 cnco 23190 . . . . . . . . . 10 (((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8614, 24, 85syl2anc 582 . . . . . . . . 9 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8717fveq2d 6906 . . . . . . . . . 10 (𝜑 → (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)) = (𝐺𝑂))
8829, 30cnf 23170 . . . . . . . . . . . 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 7002 . . . . . . . . . . 11 (((𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9189, 33, 90sylancl 584 . . . . . . . . . 10 (𝜑 → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9287, 91, 363eqtr4rd 2779 . . . . . . . . 9 (𝜑 → (𝐹𝑃) = ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0))
9321cvmlift 34942 . . . . . . . . 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 837 . . . . . . . 8 (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
95 coeq2 5865 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝐹𝑔) = (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)))
9695eqeq1d 2730 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ↔ (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
97 fveq1 6901 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝑔‘0) = ((𝑅(*𝑝𝐶)𝐼)‘0))
9897eqeq1d 2730 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃))
9996, 98anbi12d 630 . . . . . . . . 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 7408 . . . . . . . 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 582 . . . . . . 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 710 . . . . . 6 (𝜑 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*𝑝𝐶)𝐼))
103102fveq1d 6904 . . . . 5 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑅(*𝑝𝐶)𝐼)‘1))
10471, 72pco1 24962 . . . . 5 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘1) = (𝐼‘1))
105103, 104eqtrd 2768 . . . 4 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106 fveq1 6901 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘0) = ((𝑄(*𝑝𝐾)𝑁)‘0))
107106eqeq1d 2730 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂))
108 fveq1 6901 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘1) = ((𝑄(*𝑝𝐾)𝑁)‘1))
109108eqeq1d 2730 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍))
110 coeq2 5865 . . . . . . . . . . 11 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝐺𝑓) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
111110eqeq2d 2739 . . . . . . . . . 10 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
112111anbi1d 629 . . . . . . . . 9 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
113112riotabidv 7384 . . . . . . . 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 6904 . . . . . . 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 2730 . . . . . 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 1432 . . . . 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 3611 . . . 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 1369 . . 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 3983 . . . 4 (𝜑𝑍𝑌)
12121, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem4 34965 . . . 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 685 . . 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 256 . 2 (𝜑 → (𝐻𝑍) = (𝐼‘1))
124 iiconn 24827 . . . . 5 II ∈ Conn
125124a1i 11 . . . 4 (𝜑 → II ∈ Conn)
126 cvmtop1 34903 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
12723, 126syl 17 . . . . . . 7 (𝜑𝐶 ∈ Top)
12821toptopon 22839 . . . . . . 7 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129127, 128sylib 217 . . . . . 6 (𝜑𝐶 ∈ (TopOn‘𝐵))
13069rneqd 5944 . . . . . . . . 9 (𝜑 → ran (𝐹𝐼) = ran (𝐺𝑁))
131 rnco2 6262 . . . . . . . . 9 ran (𝐹𝐼) = (𝐹 “ ran 𝐼)
132 rnco2 6262 . . . . . . . . 9 ran (𝐺𝑁) = (𝐺 “ ran 𝑁)
133130, 131, 1323eqtr3g 2791 . . . . . . . 8 (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134 iitopon 24819 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
135134a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
13630toptopon 22839 . . . . . . . . . . . . . 14 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
1374, 136sylib 217 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ (TopOn‘𝑌))
138 resttopon 23085 . . . . . . . . . . . . 13 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
139137, 53, 138syl2anc 582 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
140 cnf2 23173 . . . . . . . . . . . 12 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ 𝑁 ∈ (II Cn (𝐾t 𝑀))) → 𝑁:(0[,]1)⟶𝑀)
141135, 139, 7, 140syl3anc 1368 . . . . . . . . . . 11 (𝜑𝑁:(0[,]1)⟶𝑀)
142141frnd 6735 . . . . . . . . . 10 (𝜑 → ran 𝑁𝑀)
143142, 47sstrd 3992 . . . . . . . . 9 (𝜑 → ran 𝑁 ⊆ (𝐺𝐴))
14451ffund 6731 . . . . . . . . . 10 (𝜑 → Fun 𝐺)
145143, 48sstrdi 3994 . . . . . . . . . 10 (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146 funimass3 7068 . . . . . . . . . 10 ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
147144, 145, 146syl2anc 582 . . . . . . . . 9 (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
148143, 147mpbird 256 . . . . . . . 8 (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149133, 148eqsstrd 4020 . . . . . . 7 (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
15021, 49cnf 23170 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
15177, 150syl 17 . . . . . . . . 9 (𝜑𝐹:𝐵 𝐽)
152151ffund 6731 . . . . . . . 8 (𝜑 → Fun 𝐹)
15329, 21cnf 23170 . . . . . . . . . . 11 (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
15472, 153syl 17 . . . . . . . . . 10 (𝜑𝐼:(0[,]1)⟶𝐵)
155154frnd 6735 . . . . . . . . 9 (𝜑 → ran 𝐼𝐵)
156151fdmd 6738 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐵)
157155, 156sseqtrrd 4023 . . . . . . . 8 (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158 funimass3 7068 . . . . . . . 8 ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
159152, 157, 158syl2anc 582 . . . . . . 7 (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
160149, 159mpbid 231 . . . . . 6 (𝜑 → ran 𝐼 ⊆ (𝐹𝐴))
161 cnvimass 6090 . . . . . . 7 (𝐹𝐴) ⊆ dom 𝐹
162161, 151fssdm 6747 . . . . . 6 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
163 cnrest2 23210 . . . . . 6 ((𝐶 ∈ (TopOn‘𝐵) ∧ ran 𝐼 ⊆ (𝐹𝐴) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
164129, 160, 162, 163syl3anc 1368 . . . . 5 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
16572, 164mpbid 231 . . . 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 34910 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
169166, 168syl 17 . . . . . 6 (𝜑𝑇𝐶)
170 cvmlift3lem7.1 . . . . . . . . 9 (𝜑 → (𝐺𝑋) ∈ 𝐴)
17166, 170eqeltrd 2829 . . . . . . . 8 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
172 cvmlift3lem7.w . . . . . . . . 9 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
173167, 21, 172cvmsiota 34920 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
17423, 166, 56, 171, 173syl13anc 1369 . . . . . . 7 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
175174simpld 493 . . . . . 6 (𝜑𝑊𝑇)
176169, 175sseldd 3983 . . . . 5 (𝜑𝑊𝐶)
177 elssuni 4944 . . . . . . 7 (𝑊𝑇𝑊 𝑇)
178175, 177syl 17 . . . . . 6 (𝜑𝑊 𝑇)
179167cvmsuni 34912 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇 = (𝐹𝐴))
180166, 179syl 17 . . . . . 6 (𝜑 𝑇 = (𝐹𝐴))
181178, 180sseqtrd 4022 . . . . 5 (𝜑𝑊 ⊆ (𝐹𝐴))
182167cvmsrcl 34907 . . . . . . . 8 (𝑇 ∈ (𝑆𝐴) → 𝐴𝐽)
183166, 182syl 17 . . . . . . 7 (𝜑𝐴𝐽)
184 cnima 23189 . . . . . . 7 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐶)
18577, 183, 184syl2anc 582 . . . . . 6 (𝜑 → (𝐹𝐴) ∈ 𝐶)
186 restopn2 23101 . . . . . 6 ((𝐶 ∈ Top ∧ (𝐹𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
187127, 185, 186syl2anc 582 . . . . 5 (𝜑 → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
188176, 181, 187mpbir2and 711 . . . 4 (𝜑𝑊 ∈ (𝐶t (𝐹𝐴)))
189167cvmscld 34916 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → 𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19023, 166, 175, 189syl3anc 1368 . . . 4 (𝜑𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19133a1i 11 . . . 4 (𝜑 → 0 ∈ (0[,]1))
192174simprd 494 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝑊)
19374, 192eqeltrd 2829 . . . 4 (𝜑 → (𝐼‘0) ∈ 𝑊)
19429, 125, 165, 188, 190, 191, 193conncn 23350 . . 3 (𝜑𝐼:(0[,]1)⟶𝑊)
195 1elunit 13487 . . 3 1 ∈ (0[,]1)
196 ffvelcdm 7096 . . 3 ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197194, 195, 196sylancl 584 . 2 (𝜑 → (𝐼‘1) ∈ 𝑊)
198123, 197eqeltrd 2829 1 (𝜑 → (𝐻𝑍) ∈ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  ∃!wreu 3372  {crab 3430  cdif 3946  cin 3948  wss 3949  c0 4326  𝒫 cpw 4606  {csn 4632   cuni 4912  cmpt 5235  ccnv 5681  dom cdm 5682  ran crn 5683  cres 5684  cima 5685  ccom 5686  Fun wfun 6547  wf 6549  cfv 6553  crio 7381  (class class class)co 7426  0cc0 11146  1c1 11147  [,]cicc 13367  t crest 17409  Topctop 22815  TopOnctopon 22832  Clsdccld 22940   Cn ccn 23148  Conncconn 23335  𝑛-Locally cnlly 23389  Homeochmeo 23677  IIcii 24815  *𝑝cpco 24947  PConncpconn 34862  SConncsconn 34863   CovMap ccvm 34898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224  ax-addf 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-ec 8733  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fsupp 9394  df-fi 9442  df-sup 9473  df-inf 9474  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13368  df-ico 13370  df-icc 13371  df-fz 13525  df-fzo 13668  df-fl 13797  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-sum 15673  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-starv 17255  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-unif 17263  df-hom 17264  df-cco 17265  df-rest 17411  df-topn 17412  df-0g 17430  df-gsum 17431  df-topgen 17432  df-pt 17433  df-prds 17436  df-xrs 17491  df-qtop 17496  df-imas 17497  df-xps 17499  df-mre 17573  df-mrc 17574  df-acs 17576  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-mulg 19031  df-cntz 19275  df-cmn 19744  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-cnfld 21287  df-top 22816  df-topon 22833  df-topsp 22855  df-bases 22869  df-cld 22943  df-ntr 22944  df-cls 22945  df-nei 23022  df-cn 23151  df-cnp 23152  df-cmp 23311  df-conn 23336  df-lly 23390  df-nlly 23391  df-tx 23486  df-hmeo 23679  df-xms 24246  df-ms 24247  df-tms 24248  df-ii 24817  df-cncf 24818  df-htpy 24916  df-phtpy 24917  df-phtpc 24938  df-pco 24952  df-pconn 34864  df-sconn 34865  df-cvm 34899
This theorem is referenced by:  cvmlift3lem7  34968
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