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Theorem cvmlift3lem6 31629
Description: Lemma for cvmlift3 31633. (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 31533 . . . . . . . 8 (𝐾 ∈ SConn → 𝐾 ∈ Top)
42, 3syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
5 cnrest2r 21305 . . . . . . 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 3799 . . . . 5 (𝜑𝑁 ∈ (II Cn 𝐾))
9 cvmlift3lem6.1 . . . . . . 7 (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))
109simp2d 1166 . . . . . 6 (𝜑 → (𝑄‘1) = 𝑋)
11 cvmlift3lem6.2 . . . . . . 7 (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
1211simpld 484 . . . . . 6 (𝜑 → (𝑁‘0) = 𝑋)
1310, 12eqtr4d 2843 . . . . 5 (𝜑 → (𝑄‘1) = (𝑁‘0))
141, 8, 13pcocn 23029 . . . 4 (𝜑 → (𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾))
151, 8pco0 23026 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = (𝑄‘0))
169simp1d 1165 . . . . 5 (𝜑 → (𝑄‘0) = 𝑂)
1715, 16eqtrd 2840 . . . 4 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂)
181, 8pco1 23027 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘1) = (𝑁‘1))
1911simprd 485 . . . . 5 (𝜑 → (𝑁‘1) = 𝑍)
2018, 19eqtrd 2840 . . . 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 21284 . . . . . . . . . . . 12 ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑄) ∈ (II Cn 𝐽))
261, 24, 25syl2anc 575 . . . . . . . . . . 11 (𝜑 → (𝐺𝑄) ∈ (II Cn 𝐽))
27 cvmlift3.p . . . . . . . . . . 11 (𝜑𝑃𝐵)
2816fveq2d 6412 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺𝑂))
29 iiuni 22897 . . . . . . . . . . . . . . 15 (0[,]1) = II
30 cvmlift3.y . . . . . . . . . . . . . . 15 𝑌 = 𝐾
3129, 30cnf 21264 . . . . . . . . . . . . . 14 (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
321, 31syl 17 . . . . . . . . . . . . 13 (𝜑𝑄:(0[,]1)⟶𝑌)
33 0elunit 12511 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
34 fvco3 6496 . . . . . . . . . . . . 13 ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
3532, 33, 34sylancl 576 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
36 cvmlift3.e . . . . . . . . . . . 12 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
3728, 35, 363eqtr4rd 2851 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = ((𝐺𝑄)‘0))
3821, 22, 23, 26, 27, 37cvmliftiota 31606 . . . . . . . . . 10 (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹𝑅) = (𝐺𝑄) ∧ (𝑅‘0) = 𝑃))
3938simp2d 1166 . . . . . . . . 9 (𝜑 → (𝐹𝑅) = (𝐺𝑄))
40 cvmlift3lem6.i . . . . . . . . . . 11 𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))
41 cnco 21284 . . . . . . . . . . . 12 ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑁) ∈ (II Cn 𝐽))
428, 24, 41syl2anc 575 . . . . . . . . . . 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 31626 . . . . . . . . . . . 12 (𝜑𝐻:𝑌𝐵)
47 cvmlift3lem7.3 . . . . . . . . . . . . . 14 (𝜑𝑀 ⊆ (𝐺𝐴))
48 cnvimass 5695 . . . . . . . . . . . . . . 15 (𝐺𝐴) ⊆ dom 𝐺
49 eqid 2806 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5030, 49cnf 21264 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
5124, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑌 𝐽)
5248, 51fssdm 6272 . . . . . . . . . . . . . 14 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
5347, 52sstrd 3808 . . . . . . . . . . . . 13 (𝜑𝑀𝑌)
54 cvmlift3lem6.x . . . . . . . . . . . . 13 (𝜑𝑋𝑀)
5553, 54sseldd 3799 . . . . . . . . . . . 12 (𝜑𝑋𝑌)
5646, 55ffvelrnd 6582 . . . . . . . . . . 11 (𝜑 → (𝐻𝑋) ∈ 𝐵)
5712fveq2d 6412 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺𝑋))
5829, 30cnf 21264 . . . . . . . . . . . . . 14 (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
598, 58syl 17 . . . . . . . . . . . . 13 (𝜑𝑁:(0[,]1)⟶𝑌)
60 fvco3 6496 . . . . . . . . . . . . 13 ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
6159, 33, 60sylancl 576 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
62 fvco3 6496 . . . . . . . . . . . . . 14 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6346, 55, 62syl2anc 575 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6421, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem5 31628 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐻) = 𝐺)
6564fveq1d 6410 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
6663, 65eqtr3d 2842 . . . . . . . . . . . 12 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
6757, 61, 663eqtr4rd 2851 . . . . . . . . . . 11 (𝜑 → (𝐹‘(𝐻𝑋)) = ((𝐺𝑁)‘0))
6821, 40, 23, 42, 56, 67cvmliftiota 31606 . . . . . . . . . 10 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹𝐼) = (𝐺𝑁) ∧ (𝐼‘0) = (𝐻𝑋)))
6968simp2d 1166 . . . . . . . . 9 (𝜑 → (𝐹𝐼) = (𝐺𝑁))
7039, 69oveq12d 6892 . . . . . . . 8 (𝜑 → ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
7138simp1d 1165 . . . . . . . . 9 (𝜑𝑅 ∈ (II Cn 𝐶))
7268simp1d 1165 . . . . . . . . 9 (𝜑𝐼 ∈ (II Cn 𝐶))
739simp3d 1167 . . . . . . . . . 10 (𝜑 → (𝑅‘1) = (𝐻𝑋))
7468simp3d 1167 . . . . . . . . . 10 (𝜑 → (𝐼‘0) = (𝐻𝑋))
7573, 74eqtr4d 2843 . . . . . . . . 9 (𝜑 → (𝑅‘1) = (𝐼‘0))
76 cvmcn 31567 . . . . . . . . . 10 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
7723, 76syl 17 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
7871, 72, 75, 77copco 23030 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)))
791, 8, 13, 24copco 23030 . . . . . . . 8 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
8070, 78, 793eqtr4d 2850 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
8171, 72pco0 23026 . . . . . . . 8 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = (𝑅‘0))
8238simp3d 1167 . . . . . . . 8 (𝜑 → (𝑅‘0) = 𝑃)
8381, 82eqtrd 2840 . . . . . . 7 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃)
8471, 72, 75pcocn 23029 . . . . . . . 8 (𝜑 → (𝑅(*𝑝𝐶)𝐼) ∈ (II Cn 𝐶))
85 cnco 21284 . . . . . . . . . 10 (((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8614, 24, 85syl2anc 575 . . . . . . . . 9 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8717fveq2d 6412 . . . . . . . . . 10 (𝜑 → (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)) = (𝐺𝑂))
8829, 30cnf 21264 . . . . . . . . . . . 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 6496 . . . . . . . . . . 11 (((𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9189, 33, 90sylancl 576 . . . . . . . . . 10 (𝜑 → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9287, 91, 363eqtr4rd 2851 . . . . . . . . 9 (𝜑 → (𝐹𝑃) = ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0))
9321cvmlift 31604 . . . . . . . . 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 858 . . . . . . . 8 (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
95 coeq2 5482 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝐹𝑔) = (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)))
9695eqeq1d 2808 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ↔ (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
97 fveq1 6407 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝑔‘0) = ((𝑅(*𝑝𝐶)𝐼)‘0))
9897eqeq1d 2808 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃))
9996, 98anbi12d 618 . . . . . . . . 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 6857 . . . . . . . 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 575 . . . . . . 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 694 . . . . . 6 (𝜑 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*𝑝𝐶)𝐼))
103102fveq1d 6410 . . . . 5 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑅(*𝑝𝐶)𝐼)‘1))
10471, 72pco1 23027 . . . . 5 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘1) = (𝐼‘1))
105103, 104eqtrd 2840 . . . 4 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106 fveq1 6407 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘0) = ((𝑄(*𝑝𝐾)𝑁)‘0))
107106eqeq1d 2808 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂))
108 fveq1 6407 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘1) = ((𝑄(*𝑝𝐾)𝑁)‘1))
109108eqeq1d 2808 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍))
110 coeq2 5482 . . . . . . . . . . 11 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝐺𝑓) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
111110eqeq2d 2816 . . . . . . . . . 10 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
112111anbi1d 617 . . . . . . . . 9 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
113112riotabidv 6837 . . . . . . . 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 6410 . . . . . . 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 2808 . . . . . 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 1553 . . . . 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 3502 . . . 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 1484 . . 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 3799 . . . 4 (𝜑𝑍𝑌)
12121, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem4 31627 . . . 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 670 . . 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 248 . 2 (𝜑 → (𝐻𝑍) = (𝐼‘1))
124 iiconn 22903 . . . . 5 II ∈ Conn
125124a1i 11 . . . 4 (𝜑 → II ∈ Conn)
126 cvmtop1 31565 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
12723, 126syl 17 . . . . . . 7 (𝜑𝐶 ∈ Top)
12821toptopon 20935 . . . . . . 7 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129127, 128sylib 209 . . . . . 6 (𝜑𝐶 ∈ (TopOn‘𝐵))
13069rneqd 5554 . . . . . . . . 9 (𝜑 → ran (𝐹𝐼) = ran (𝐺𝑁))
131 rnco2 5856 . . . . . . . . 9 ran (𝐹𝐼) = (𝐹 “ ran 𝐼)
132 rnco2 5856 . . . . . . . . 9 ran (𝐺𝑁) = (𝐺 “ ran 𝑁)
133130, 131, 1323eqtr3g 2863 . . . . . . . 8 (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134 iitopon 22895 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
135134a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
13630toptopon 20935 . . . . . . . . . . . . . 14 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
1374, 136sylib 209 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ (TopOn‘𝑌))
138 resttopon 21179 . . . . . . . . . . . . 13 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
139137, 53, 138syl2anc 575 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
140 cnf2 21267 . . . . . . . . . . . 12 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ 𝑁 ∈ (II Cn (𝐾t 𝑀))) → 𝑁:(0[,]1)⟶𝑀)
141135, 139, 7, 140syl3anc 1483 . . . . . . . . . . 11 (𝜑𝑁:(0[,]1)⟶𝑀)
142141frnd 6263 . . . . . . . . . 10 (𝜑 → ran 𝑁𝑀)
143142, 47sstrd 3808 . . . . . . . . 9 (𝜑 → ran 𝑁 ⊆ (𝐺𝐴))
14451ffund 6260 . . . . . . . . . 10 (𝜑 → Fun 𝐺)
145143, 48syl6ss 3810 . . . . . . . . . 10 (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146 funimass3 6555 . . . . . . . . . 10 ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
147144, 145, 146syl2anc 575 . . . . . . . . 9 (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
148143, 147mpbird 248 . . . . . . . 8 (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149133, 148eqsstrd 3836 . . . . . . 7 (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
15021, 49cnf 21264 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
15177, 150syl 17 . . . . . . . . 9 (𝜑𝐹:𝐵 𝐽)
152151ffund 6260 . . . . . . . 8 (𝜑 → Fun 𝐹)
15329, 21cnf 21264 . . . . . . . . . . 11 (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
15472, 153syl 17 . . . . . . . . . 10 (𝜑𝐼:(0[,]1)⟶𝐵)
155154frnd 6263 . . . . . . . . 9 (𝜑 → ran 𝐼𝐵)
156151fdmd 6265 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐵)
157155, 156sseqtr4d 3839 . . . . . . . 8 (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158 funimass3 6555 . . . . . . . 8 ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
159152, 157, 158syl2anc 575 . . . . . . 7 (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
160149, 159mpbid 223 . . . . . 6 (𝜑 → ran 𝐼 ⊆ (𝐹𝐴))
161 cnvimass 5695 . . . . . . 7 (𝐹𝐴) ⊆ dom 𝐹
162161, 151fssdm 6272 . . . . . 6 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
163 cnrest2 21304 . . . . . 6 ((𝐶 ∈ (TopOn‘𝐵) ∧ ran 𝐼 ⊆ (𝐹𝐴) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
164129, 160, 162, 163syl3anc 1483 . . . . 5 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
16572, 164mpbid 223 . . . 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 31572 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
169166, 168syl 17 . . . . . 6 (𝜑𝑇𝐶)
170 cvmlift3lem7.1 . . . . . . . . 9 (𝜑 → (𝐺𝑋) ∈ 𝐴)
17166, 170eqeltrd 2885 . . . . . . . 8 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
172 cvmlift3lem7.w . . . . . . . . 9 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
173167, 21, 172cvmsiota 31582 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
17423, 166, 56, 171, 173syl13anc 1484 . . . . . . 7 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
175174simpld 484 . . . . . 6 (𝜑𝑊𝑇)
176169, 175sseldd 3799 . . . . 5 (𝜑𝑊𝐶)
177 elssuni 4661 . . . . . . 7 (𝑊𝑇𝑊 𝑇)
178175, 177syl 17 . . . . . 6 (𝜑𝑊 𝑇)
179167cvmsuni 31574 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇 = (𝐹𝐴))
180166, 179syl 17 . . . . . 6 (𝜑 𝑇 = (𝐹𝐴))
181178, 180sseqtrd 3838 . . . . 5 (𝜑𝑊 ⊆ (𝐹𝐴))
182167cvmsrcl 31569 . . . . . . . 8 (𝑇 ∈ (𝑆𝐴) → 𝐴𝐽)
183166, 182syl 17 . . . . . . 7 (𝜑𝐴𝐽)
184 cnima 21283 . . . . . . 7 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐶)
18577, 183, 184syl2anc 575 . . . . . 6 (𝜑 → (𝐹𝐴) ∈ 𝐶)
186 restopn2 21195 . . . . . 6 ((𝐶 ∈ Top ∧ (𝐹𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
187127, 185, 186syl2anc 575 . . . . 5 (𝜑 → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
188176, 181, 187mpbir2and 695 . . . 4 (𝜑𝑊 ∈ (𝐶t (𝐹𝐴)))
189167cvmscld 31578 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → 𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19023, 166, 175, 189syl3anc 1483 . . . 4 (𝜑𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19133a1i 11 . . . 4 (𝜑 → 0 ∈ (0[,]1))
192174simprd 485 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝑊)
19374, 192eqeltrd 2885 . . . 4 (𝜑 → (𝐼‘0) ∈ 𝑊)
19429, 125, 165, 188, 190, 191, 193conncn 21443 . . 3 (𝜑𝐼:(0[,]1)⟶𝑊)
195 1elunit 12512 . . 3 1 ∈ (0[,]1)
196 ffvelrn 6579 . . 3 ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197194, 195, 196sylancl 576 . 2 (𝜑 → (𝐼‘1) ∈ 𝑊)
198123, 197eqeltrd 2885 1 (𝜑 → (𝐻𝑍) ∈ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wral 3096  wrex 3097  ∃!wreu 3098  {crab 3100  cdif 3766  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  {csn 4370   cuni 4630  cmpt 4923  ccnv 5310  dom cdm 5311  ran crn 5312  cres 5313  cima 5314  ccom 5315  Fun wfun 6095  wf 6097  cfv 6101  crio 6834  (class class class)co 6874  0cc0 10221  1c1 10222  [,]cicc 12396  t crest 16286  Topctop 20911  TopOnctopon 20928  Clsdccld 21034   Cn ccn 21242  Conncconn 21428  𝑛-Locally cnlly 21482  Homeochmeo 21770  IIcii 22891  *𝑝cpco 23012  PConncpconn 31524  SConncsconn 31525   CovMap ccvm 31560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-om 7296  df-1st 7398  df-2nd 7399  df-supp 7530  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-ec 7981  df-map 8094  df-ixp 8146  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fsupp 8515  df-fi 8556  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-q 12008  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-ioo 12397  df-ico 12399  df-icc 12400  df-fz 12550  df-fzo 12690  df-fl 12817  df-seq 13025  df-exp 13084  df-hash 13338  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-clim 14442  df-sum 14640  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-sca 16169  df-vsca 16170  df-ip 16171  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-hom 16177  df-cco 16178  df-rest 16288  df-topn 16289  df-0g 16307  df-gsum 16308  df-topgen 16309  df-pt 16310  df-prds 16313  df-xrs 16367  df-qtop 16372  df-imas 16373  df-xps 16375  df-mre 16451  df-mrc 16452  df-acs 16454  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-submnd 17541  df-mulg 17746  df-cntz 17951  df-cmn 18396  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-cnfld 19955  df-top 20912  df-topon 20929  df-topsp 20951  df-bases 20964  df-cld 21037  df-ntr 21038  df-cls 21039  df-nei 21116  df-cn 21245  df-cnp 21246  df-cmp 21404  df-conn 21429  df-lly 21483  df-nlly 21484  df-tx 21579  df-hmeo 21772  df-xms 22338  df-ms 22339  df-tms 22340  df-ii 22893  df-htpy 22982  df-phtpy 22983  df-phtpc 23004  df-pco 23017  df-pconn 31526  df-sconn 31527  df-cvm 31561
This theorem is referenced by:  cvmlift3lem7  31630
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