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Theorem cvmlift3lem6 32579
Description: Lemma for cvmlift3 32583. (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 32483 . . . . . . . 8 (𝐾 ∈ SConn → 𝐾 ∈ Top)
42, 3syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
5 cnrest2r 21871 . . . . . . 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 3947 . . . . 5 (𝜑𝑁 ∈ (II Cn 𝐾))
9 cvmlift3lem6.1 . . . . . . 7 (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))
109simp2d 1139 . . . . . 6 (𝜑 → (𝑄‘1) = 𝑋)
11 cvmlift3lem6.2 . . . . . . 7 (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
1211simpld 497 . . . . . 6 (𝜑 → (𝑁‘0) = 𝑋)
1310, 12eqtr4d 2858 . . . . 5 (𝜑 → (𝑄‘1) = (𝑁‘0))
141, 8, 13pcocn 23601 . . . 4 (𝜑 → (𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾))
151, 8pco0 23598 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = (𝑄‘0))
169simp1d 1138 . . . . 5 (𝜑 → (𝑄‘0) = 𝑂)
1715, 16eqtrd 2855 . . . 4 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂)
181, 8pco1 23599 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘1) = (𝑁‘1))
1911simprd 498 . . . . 5 (𝜑 → (𝑁‘1) = 𝑍)
2018, 19eqtrd 2855 . . . 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 21850 . . . . . . . . . . . 12 ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑄) ∈ (II Cn 𝐽))
261, 24, 25syl2anc 586 . . . . . . . . . . 11 (𝜑 → (𝐺𝑄) ∈ (II Cn 𝐽))
27 cvmlift3.p . . . . . . . . . . 11 (𝜑𝑃𝐵)
2816fveq2d 6650 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺𝑂))
29 iiuni 23465 . . . . . . . . . . . . . . 15 (0[,]1) = II
30 cvmlift3.y . . . . . . . . . . . . . . 15 𝑌 = 𝐾
3129, 30cnf 21830 . . . . . . . . . . . . . 14 (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
321, 31syl 17 . . . . . . . . . . . . 13 (𝜑𝑄:(0[,]1)⟶𝑌)
33 0elunit 12838 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
34 fvco3 6736 . . . . . . . . . . . . 13 ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
3532, 33, 34sylancl 588 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
36 cvmlift3.e . . . . . . . . . . . 12 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
3728, 35, 363eqtr4rd 2866 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = ((𝐺𝑄)‘0))
3821, 22, 23, 26, 27, 37cvmliftiota 32556 . . . . . . . . . 10 (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹𝑅) = (𝐺𝑄) ∧ (𝑅‘0) = 𝑃))
3938simp2d 1139 . . . . . . . . 9 (𝜑 → (𝐹𝑅) = (𝐺𝑄))
40 cvmlift3lem6.i . . . . . . . . . . 11 𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))
41 cnco 21850 . . . . . . . . . . . 12 ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑁) ∈ (II Cn 𝐽))
428, 24, 41syl2anc 586 . . . . . . . . . . 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 32576 . . . . . . . . . . . 12 (𝜑𝐻:𝑌𝐵)
47 cvmlift3lem7.3 . . . . . . . . . . . . . 14 (𝜑𝑀 ⊆ (𝐺𝐴))
48 cnvimass 5925 . . . . . . . . . . . . . . 15 (𝐺𝐴) ⊆ dom 𝐺
49 eqid 2820 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5030, 49cnf 21830 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
5124, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑌 𝐽)
5248, 51fssdm 6506 . . . . . . . . . . . . . 14 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
5347, 52sstrd 3956 . . . . . . . . . . . . 13 (𝜑𝑀𝑌)
54 cvmlift3lem6.x . . . . . . . . . . . . 13 (𝜑𝑋𝑀)
5553, 54sseldd 3947 . . . . . . . . . . . 12 (𝜑𝑋𝑌)
5646, 55ffvelrnd 6828 . . . . . . . . . . 11 (𝜑 → (𝐻𝑋) ∈ 𝐵)
5712fveq2d 6650 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺𝑋))
5829, 30cnf 21830 . . . . . . . . . . . . . 14 (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
598, 58syl 17 . . . . . . . . . . . . 13 (𝜑𝑁:(0[,]1)⟶𝑌)
60 fvco3 6736 . . . . . . . . . . . . 13 ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
6159, 33, 60sylancl 588 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
62 fvco3 6736 . . . . . . . . . . . . . 14 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6346, 55, 62syl2anc 586 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6421, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem5 32578 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐻) = 𝐺)
6564fveq1d 6648 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
6663, 65eqtr3d 2857 . . . . . . . . . . . 12 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
6757, 61, 663eqtr4rd 2866 . . . . . . . . . . 11 (𝜑 → (𝐹‘(𝐻𝑋)) = ((𝐺𝑁)‘0))
6821, 40, 23, 42, 56, 67cvmliftiota 32556 . . . . . . . . . 10 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹𝐼) = (𝐺𝑁) ∧ (𝐼‘0) = (𝐻𝑋)))
6968simp2d 1139 . . . . . . . . 9 (𝜑 → (𝐹𝐼) = (𝐺𝑁))
7039, 69oveq12d 7151 . . . . . . . 8 (𝜑 → ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
7138simp1d 1138 . . . . . . . . 9 (𝜑𝑅 ∈ (II Cn 𝐶))
7268simp1d 1138 . . . . . . . . 9 (𝜑𝐼 ∈ (II Cn 𝐶))
739simp3d 1140 . . . . . . . . . 10 (𝜑 → (𝑅‘1) = (𝐻𝑋))
7468simp3d 1140 . . . . . . . . . 10 (𝜑 → (𝐼‘0) = (𝐻𝑋))
7573, 74eqtr4d 2858 . . . . . . . . 9 (𝜑 → (𝑅‘1) = (𝐼‘0))
76 cvmcn 32517 . . . . . . . . . 10 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
7723, 76syl 17 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
7871, 72, 75, 77copco 23602 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)))
791, 8, 13, 24copco 23602 . . . . . . . 8 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
8070, 78, 793eqtr4d 2865 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
8171, 72pco0 23598 . . . . . . . 8 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = (𝑅‘0))
8238simp3d 1140 . . . . . . . 8 (𝜑 → (𝑅‘0) = 𝑃)
8381, 82eqtrd 2855 . . . . . . 7 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃)
8471, 72, 75pcocn 23601 . . . . . . . 8 (𝜑 → (𝑅(*𝑝𝐶)𝐼) ∈ (II Cn 𝐶))
85 cnco 21850 . . . . . . . . . 10 (((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8614, 24, 85syl2anc 586 . . . . . . . . 9 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8717fveq2d 6650 . . . . . . . . . 10 (𝜑 → (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)) = (𝐺𝑂))
8829, 30cnf 21830 . . . . . . . . . . . 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 6736 . . . . . . . . . . 11 (((𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9189, 33, 90sylancl 588 . . . . . . . . . 10 (𝜑 → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9287, 91, 363eqtr4rd 2866 . . . . . . . . 9 (𝜑 → (𝐹𝑃) = ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0))
9321cvmlift 32554 . . . . . . . . 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 836 . . . . . . . 8 (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
95 coeq2 5705 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝐹𝑔) = (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)))
9695eqeq1d 2822 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ↔ (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
97 fveq1 6645 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝑔‘0) = ((𝑅(*𝑝𝐶)𝐼)‘0))
9897eqeq1d 2822 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃))
9996, 98anbi12d 632 . . . . . . . . 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 7116 . . . . . . . 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 586 . . . . . . 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 6648 . . . . 5 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑅(*𝑝𝐶)𝐼)‘1))
10471, 72pco1 23599 . . . . 5 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘1) = (𝐼‘1))
105103, 104eqtrd 2855 . . . 4 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106 fveq1 6645 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘0) = ((𝑄(*𝑝𝐾)𝑁)‘0))
107106eqeq1d 2822 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂))
108 fveq1 6645 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘1) = ((𝑄(*𝑝𝐾)𝑁)‘1))
109108eqeq1d 2822 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍))
110 coeq2 5705 . . . . . . . . . . 11 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝐺𝑓) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
111110eqeq2d 2831 . . . . . . . . . 10 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
112111anbi1d 631 . . . . . . . . 9 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
113112riotabidv 7093 . . . . . . . 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 6648 . . . . . . 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 2822 . . . . . 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 3602 . . . 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 1368 . . 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 3947 . . . 4 (𝜑𝑍𝑌)
12121, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem4 32577 . . . 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 259 . 2 (𝜑 → (𝐻𝑍) = (𝐼‘1))
124 iiconn 23471 . . . . 5 II ∈ Conn
125124a1i 11 . . . 4 (𝜑 → II ∈ Conn)
126 cvmtop1 32515 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
12723, 126syl 17 . . . . . . 7 (𝜑𝐶 ∈ Top)
12821toptopon 21501 . . . . . . 7 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129127, 128sylib 220 . . . . . 6 (𝜑𝐶 ∈ (TopOn‘𝐵))
13069rneqd 5784 . . . . . . . . 9 (𝜑 → ran (𝐹𝐼) = ran (𝐺𝑁))
131 rnco2 6082 . . . . . . . . 9 ran (𝐹𝐼) = (𝐹 “ ran 𝐼)
132 rnco2 6082 . . . . . . . . 9 ran (𝐺𝑁) = (𝐺 “ ran 𝑁)
133130, 131, 1323eqtr3g 2878 . . . . . . . 8 (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134 iitopon 23463 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
135134a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
13630toptopon 21501 . . . . . . . . . . . . . 14 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
1374, 136sylib 220 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ (TopOn‘𝑌))
138 resttopon 21745 . . . . . . . . . . . . 13 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
139137, 53, 138syl2anc 586 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
140 cnf2 21833 . . . . . . . . . . . 12 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ 𝑁 ∈ (II Cn (𝐾t 𝑀))) → 𝑁:(0[,]1)⟶𝑀)
141135, 139, 7, 140syl3anc 1367 . . . . . . . . . . 11 (𝜑𝑁:(0[,]1)⟶𝑀)
142141frnd 6497 . . . . . . . . . 10 (𝜑 → ran 𝑁𝑀)
143142, 47sstrd 3956 . . . . . . . . 9 (𝜑 → ran 𝑁 ⊆ (𝐺𝐴))
14451ffund 6494 . . . . . . . . . 10 (𝜑 → Fun 𝐺)
145143, 48sstrdi 3958 . . . . . . . . . 10 (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146 funimass3 6800 . . . . . . . . . 10 ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
147144, 145, 146syl2anc 586 . . . . . . . . 9 (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
148143, 147mpbird 259 . . . . . . . 8 (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149133, 148eqsstrd 3984 . . . . . . 7 (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
15021, 49cnf 21830 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
15177, 150syl 17 . . . . . . . . 9 (𝜑𝐹:𝐵 𝐽)
152151ffund 6494 . . . . . . . 8 (𝜑 → Fun 𝐹)
15329, 21cnf 21830 . . . . . . . . . . 11 (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
15472, 153syl 17 . . . . . . . . . 10 (𝜑𝐼:(0[,]1)⟶𝐵)
155154frnd 6497 . . . . . . . . 9 (𝜑 → ran 𝐼𝐵)
156151fdmd 6499 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐵)
157155, 156sseqtrrd 3987 . . . . . . . 8 (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158 funimass3 6800 . . . . . . . 8 ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
159152, 157, 158syl2anc 586 . . . . . . 7 (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
160149, 159mpbid 234 . . . . . 6 (𝜑 → ran 𝐼 ⊆ (𝐹𝐴))
161 cnvimass 5925 . . . . . . 7 (𝐹𝐴) ⊆ dom 𝐹
162161, 151fssdm 6506 . . . . . 6 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
163 cnrest2 21870 . . . . . 6 ((𝐶 ∈ (TopOn‘𝐵) ∧ ran 𝐼 ⊆ (𝐹𝐴) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
164129, 160, 162, 163syl3anc 1367 . . . . 5 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
16572, 164mpbid 234 . . . 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 32522 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
169166, 168syl 17 . . . . . 6 (𝜑𝑇𝐶)
170 cvmlift3lem7.1 . . . . . . . . 9 (𝜑 → (𝐺𝑋) ∈ 𝐴)
17166, 170eqeltrd 2911 . . . . . . . 8 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
172 cvmlift3lem7.w . . . . . . . . 9 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
173167, 21, 172cvmsiota 32532 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
17423, 166, 56, 171, 173syl13anc 1368 . . . . . . 7 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
175174simpld 497 . . . . . 6 (𝜑𝑊𝑇)
176169, 175sseldd 3947 . . . . 5 (𝜑𝑊𝐶)
177 elssuni 4844 . . . . . . 7 (𝑊𝑇𝑊 𝑇)
178175, 177syl 17 . . . . . 6 (𝜑𝑊 𝑇)
179167cvmsuni 32524 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇 = (𝐹𝐴))
180166, 179syl 17 . . . . . 6 (𝜑 𝑇 = (𝐹𝐴))
181178, 180sseqtrd 3986 . . . . 5 (𝜑𝑊 ⊆ (𝐹𝐴))
182167cvmsrcl 32519 . . . . . . . 8 (𝑇 ∈ (𝑆𝐴) → 𝐴𝐽)
183166, 182syl 17 . . . . . . 7 (𝜑𝐴𝐽)
184 cnima 21849 . . . . . . 7 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐶)
18577, 183, 184syl2anc 586 . . . . . 6 (𝜑 → (𝐹𝐴) ∈ 𝐶)
186 restopn2 21761 . . . . . 6 ((𝐶 ∈ Top ∧ (𝐹𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
187127, 185, 186syl2anc 586 . . . . 5 (𝜑 → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
188176, 181, 187mpbir2and 711 . . . 4 (𝜑𝑊 ∈ (𝐶t (𝐹𝐴)))
189167cvmscld 32528 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → 𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19023, 166, 175, 189syl3anc 1367 . . . 4 (𝜑𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19133a1i 11 . . . 4 (𝜑 → 0 ∈ (0[,]1))
192174simprd 498 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝑊)
19374, 192eqeltrd 2911 . . . 4 (𝜑 → (𝐼‘0) ∈ 𝑊)
19429, 125, 165, 188, 190, 191, 193conncn 22010 . . 3 (𝜑𝐼:(0[,]1)⟶𝑊)
195 1elunit 12839 . . 3 1 ∈ (0[,]1)
196 ffvelrn 6825 . . 3 ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197194, 195, 196sylancl 588 . 2 (𝜑 → (𝐼‘1) ∈ 𝑊)
198123, 197eqeltrd 2911 1 (𝜑 → (𝐻𝑍) ∈ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  wrex 3126  ∃!wreu 3127  {crab 3129  cdif 3910  cin 3912  wss 3913  c0 4269  𝒫 cpw 4515  {csn 4543   cuni 4814  cmpt 5122  ccnv 5530  dom cdm 5531  ran crn 5532  cres 5533  cima 5534  ccom 5535  Fun wfun 6325  wf 6327  cfv 6331  crio 7090  (class class class)co 7133  0cc0 10515  1c1 10516  [,]cicc 12720  t crest 16673  Topctop 21477  TopOnctopon 21494  Clsdccld 21600   Cn ccn 21808  Conncconn 21995  𝑛-Locally cnlly 22049  Homeochmeo 22337  IIcii 23459  *𝑝cpco 23584  PConncpconn 32474  SConncsconn 32475   CovMap ccvm 32510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593  ax-addf 10594  ax-mulf 10595
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-ec 8269  df-map 8386  df-ixp 8440  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-fsupp 8812  df-fi 8853  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-q 12328  df-rp 12369  df-xneg 12486  df-xadd 12487  df-xmul 12488  df-ioo 12721  df-ico 12723  df-icc 12724  df-fz 12877  df-fzo 13018  df-fl 13146  df-seq 13354  df-exp 13415  df-hash 13676  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-clim 14825  df-sum 15023  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-mulr 16558  df-starv 16559  df-sca 16560  df-vsca 16561  df-ip 16562  df-tset 16563  df-ple 16564  df-ds 16566  df-unif 16567  df-hom 16568  df-cco 16569  df-rest 16675  df-topn 16676  df-0g 16694  df-gsum 16695  df-topgen 16696  df-pt 16697  df-prds 16700  df-xrs 16754  df-qtop 16759  df-imas 16760  df-xps 16762  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-mulg 18204  df-cntz 18426  df-cmn 18887  df-psmet 20513  df-xmet 20514  df-met 20515  df-bl 20516  df-mopn 20517  df-cnfld 20522  df-top 21478  df-topon 21495  df-topsp 21517  df-bases 21530  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-cn 21811  df-cnp 21812  df-cmp 21971  df-conn 21996  df-lly 22050  df-nlly 22051  df-tx 22146  df-hmeo 22339  df-xms 22906  df-ms 22907  df-tms 22908  df-ii 23461  df-htpy 23554  df-phtpy 23555  df-phtpc 23576  df-pco 23589  df-pconn 32476  df-sconn 32477  df-cvm 32511
This theorem is referenced by:  cvmlift3lem7  32580
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