Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3lem6 Structured version   Visualization version   GIF version

Theorem cvmlift3lem6 33918
Description: Lemma for cvmlift3 33922. (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 33822 . . . . . . . 8 (𝐾 ∈ SConn → 𝐾 ∈ Top)
42, 3syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
5 cnrest2r 22638 . . . . . . 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 3945 . . . . 5 (𝜑𝑁 ∈ (II Cn 𝐾))
9 cvmlift3lem6.1 . . . . . . 7 (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))
109simp2d 1143 . . . . . 6 (𝜑 → (𝑄‘1) = 𝑋)
11 cvmlift3lem6.2 . . . . . . 7 (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
1211simpld 495 . . . . . 6 (𝜑 → (𝑁‘0) = 𝑋)
1310, 12eqtr4d 2779 . . . . 5 (𝜑 → (𝑄‘1) = (𝑁‘0))
141, 8, 13pcocn 24380 . . . 4 (𝜑 → (𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾))
151, 8pco0 24377 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = (𝑄‘0))
169simp1d 1142 . . . . 5 (𝜑 → (𝑄‘0) = 𝑂)
1715, 16eqtrd 2776 . . . 4 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂)
181, 8pco1 24378 . . . . 5 (𝜑 → ((𝑄(*𝑝𝐾)𝑁)‘1) = (𝑁‘1))
1911simprd 496 . . . . 5 (𝜑 → (𝑁‘1) = 𝑍)
2018, 19eqtrd 2776 . . . 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 22617 . . . . . . . . . . . 12 ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑄) ∈ (II Cn 𝐽))
261, 24, 25syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐺𝑄) ∈ (II Cn 𝐽))
27 cvmlift3.p . . . . . . . . . . 11 (𝜑𝑃𝐵)
2816fveq2d 6846 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺𝑂))
29 iiuni 24244 . . . . . . . . . . . . . . 15 (0[,]1) = II
30 cvmlift3.y . . . . . . . . . . . . . . 15 𝑌 = 𝐾
3129, 30cnf 22597 . . . . . . . . . . . . . 14 (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
321, 31syl 17 . . . . . . . . . . . . 13 (𝜑𝑄:(0[,]1)⟶𝑌)
33 0elunit 13386 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
34 fvco3 6940 . . . . . . . . . . . . 13 ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
3532, 33, 34sylancl 586 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑄)‘0) = (𝐺‘(𝑄‘0)))
36 cvmlift3.e . . . . . . . . . . . 12 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
3728, 35, 363eqtr4rd 2787 . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = ((𝐺𝑄)‘0))
3821, 22, 23, 26, 27, 37cvmliftiota 33895 . . . . . . . . . 10 (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹𝑅) = (𝐺𝑄) ∧ (𝑅‘0) = 𝑃))
3938simp2d 1143 . . . . . . . . 9 (𝜑 → (𝐹𝑅) = (𝐺𝑄))
40 cvmlift3lem6.i . . . . . . . . . . 11 𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))
41 cnco 22617 . . . . . . . . . . . 12 ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑁) ∈ (II Cn 𝐽))
428, 24, 41syl2anc 584 . . . . . . . . . . 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 33915 . . . . . . . . . . . 12 (𝜑𝐻:𝑌𝐵)
47 cvmlift3lem7.3 . . . . . . . . . . . . . 14 (𝜑𝑀 ⊆ (𝐺𝐴))
48 cnvimass 6033 . . . . . . . . . . . . . . 15 (𝐺𝐴) ⊆ dom 𝐺
49 eqid 2736 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5030, 49cnf 22597 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
5124, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝑌 𝐽)
5248, 51fssdm 6688 . . . . . . . . . . . . . 14 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
5347, 52sstrd 3954 . . . . . . . . . . . . 13 (𝜑𝑀𝑌)
54 cvmlift3lem6.x . . . . . . . . . . . . 13 (𝜑𝑋𝑀)
5553, 54sseldd 3945 . . . . . . . . . . . 12 (𝜑𝑋𝑌)
5646, 55ffvelcdmd 7036 . . . . . . . . . . 11 (𝜑 → (𝐻𝑋) ∈ 𝐵)
5712fveq2d 6846 . . . . . . . . . . . 12 (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺𝑋))
5829, 30cnf 22597 . . . . . . . . . . . . . 14 (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
598, 58syl 17 . . . . . . . . . . . . 13 (𝜑𝑁:(0[,]1)⟶𝑌)
60 fvco3 6940 . . . . . . . . . . . . 13 ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
6159, 33, 60sylancl 586 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑁)‘0) = (𝐺‘(𝑁‘0)))
62 fvco3 6940 . . . . . . . . . . . . . 14 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6346, 55, 62syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
6421, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem5 33917 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝐻) = 𝐺)
6564fveq1d 6844 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
6663, 65eqtr3d 2778 . . . . . . . . . . . 12 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
6757, 61, 663eqtr4rd 2787 . . . . . . . . . . 11 (𝜑 → (𝐹‘(𝐻𝑋)) = ((𝐺𝑁)‘0))
6821, 40, 23, 42, 56, 67cvmliftiota 33895 . . . . . . . . . 10 (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹𝐼) = (𝐺𝑁) ∧ (𝐼‘0) = (𝐻𝑋)))
6968simp2d 1143 . . . . . . . . 9 (𝜑 → (𝐹𝐼) = (𝐺𝑁))
7039, 69oveq12d 7375 . . . . . . . 8 (𝜑 → ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
7138simp1d 1142 . . . . . . . . 9 (𝜑𝑅 ∈ (II Cn 𝐶))
7268simp1d 1142 . . . . . . . . 9 (𝜑𝐼 ∈ (II Cn 𝐶))
739simp3d 1144 . . . . . . . . . 10 (𝜑 → (𝑅‘1) = (𝐻𝑋))
7468simp3d 1144 . . . . . . . . . 10 (𝜑 → (𝐼‘0) = (𝐻𝑋))
7573, 74eqtr4d 2779 . . . . . . . . 9 (𝜑 → (𝑅‘1) = (𝐼‘0))
76 cvmcn 33856 . . . . . . . . . 10 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
7723, 76syl 17 . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
7871, 72, 75, 77copco 24381 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = ((𝐹𝑅)(*𝑝𝐽)(𝐹𝐼)))
791, 8, 13, 24copco 24381 . . . . . . . 8 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) = ((𝐺𝑄)(*𝑝𝐽)(𝐺𝑁)))
8070, 78, 793eqtr4d 2786 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
8171, 72pco0 24377 . . . . . . . 8 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = (𝑅‘0))
8238simp3d 1144 . . . . . . . 8 (𝜑 → (𝑅‘0) = 𝑃)
8381, 82eqtrd 2776 . . . . . . 7 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃)
8471, 72, 75pcocn 24380 . . . . . . . 8 (𝜑 → (𝑅(*𝑝𝐶)𝐼) ∈ (II Cn 𝐶))
85 cnco 22617 . . . . . . . . . 10 (((𝑄(*𝑝𝐾)𝑁) ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8614, 24, 85syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∈ (II Cn 𝐽))
8717fveq2d 6846 . . . . . . . . . 10 (𝜑 → (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)) = (𝐺𝑂))
8829, 30cnf 22597 . . . . . . . . . . . 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 6940 . . . . . . . . . . 11 (((𝑄(*𝑝𝐾)𝑁):(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9189, 33, 90sylancl 586 . . . . . . . . . 10 (𝜑 → ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0) = (𝐺‘((𝑄(*𝑝𝐾)𝑁)‘0)))
9287, 91, 363eqtr4rd 2787 . . . . . . . . 9 (𝜑 → (𝐹𝑃) = ((𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))‘0))
9321cvmlift 33893 . . . . . . . . 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 5814 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝐹𝑔) = (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)))
9695eqeq1d 2738 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ↔ (𝐹 ∘ (𝑅(*𝑝𝐶)𝐼)) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
97 fveq1 6841 . . . . . . . . . . 11 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → (𝑔‘0) = ((𝑅(*𝑝𝐶)𝐼)‘0))
9897eqeq1d 2738 . . . . . . . . . 10 (𝑔 = (𝑅(*𝑝𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(*𝑝𝐶)𝐼)‘0) = 𝑃))
9996, 98anbi12d 631 . . . . . . . . 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 7339 . . . . . . . 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 584 . . . . . . 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 6844 . . . . 5 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑅(*𝑝𝐶)𝐼)‘1))
10471, 72pco1 24378 . . . . 5 (𝜑 → ((𝑅(*𝑝𝐶)𝐼)‘1) = (𝐼‘1))
105103, 104eqtrd 2776 . . . 4 (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106 fveq1 6841 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘0) = ((𝑄(*𝑝𝐾)𝑁)‘0))
107106eqeq1d 2738 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(*𝑝𝐾)𝑁)‘0) = 𝑂))
108 fveq1 6841 . . . . . . 7 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝑓‘1) = ((𝑄(*𝑝𝐾)𝑁)‘1))
109108eqeq1d 2738 . . . . . 6 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(*𝑝𝐾)𝑁)‘1) = 𝑍))
110 coeq2 5814 . . . . . . . . . . 11 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (𝐺𝑓) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)))
111110eqeq2d 2747 . . . . . . . . . 10 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁))))
112111anbi1d 630 . . . . . . . . 9 (𝑓 = (𝑄(*𝑝𝐾)𝑁) → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺 ∘ (𝑄(*𝑝𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
113112riotabidv 7315 . . . . . . . 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 6844 . . . . . . 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 2738 . . . . . 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 1436 . . . . 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 3581 . . . 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 1372 . . 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 3945 . . . 4 (𝜑𝑍𝑌)
12121, 30, 23, 2, 43, 44, 24, 27, 36, 45cvmlift3lem4 33916 . . . 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 24250 . . . . 5 II ∈ Conn
125124a1i 11 . . . 4 (𝜑 → II ∈ Conn)
126 cvmtop1 33854 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
12723, 126syl 17 . . . . . . 7 (𝜑𝐶 ∈ Top)
12821toptopon 22266 . . . . . . 7 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129127, 128sylib 217 . . . . . 6 (𝜑𝐶 ∈ (TopOn‘𝐵))
13069rneqd 5893 . . . . . . . . 9 (𝜑 → ran (𝐹𝐼) = ran (𝐺𝑁))
131 rnco2 6205 . . . . . . . . 9 ran (𝐹𝐼) = (𝐹 “ ran 𝐼)
132 rnco2 6205 . . . . . . . . 9 ran (𝐺𝑁) = (𝐺 “ ran 𝑁)
133130, 131, 1323eqtr3g 2799 . . . . . . . 8 (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134 iitopon 24242 . . . . . . . . . . . . 13 II ∈ (TopOn‘(0[,]1))
135134a1i 11 . . . . . . . . . . . 12 (𝜑 → II ∈ (TopOn‘(0[,]1)))
13630toptopon 22266 . . . . . . . . . . . . . 14 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
1374, 136sylib 217 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ (TopOn‘𝑌))
138 resttopon 22512 . . . . . . . . . . . . 13 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀𝑌) → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
139137, 53, 138syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐾t 𝑀) ∈ (TopOn‘𝑀))
140 cnf2 22600 . . . . . . . . . . . 12 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝐾t 𝑀) ∈ (TopOn‘𝑀) ∧ 𝑁 ∈ (II Cn (𝐾t 𝑀))) → 𝑁:(0[,]1)⟶𝑀)
141135, 139, 7, 140syl3anc 1371 . . . . . . . . . . 11 (𝜑𝑁:(0[,]1)⟶𝑀)
142141frnd 6676 . . . . . . . . . 10 (𝜑 → ran 𝑁𝑀)
143142, 47sstrd 3954 . . . . . . . . 9 (𝜑 → ran 𝑁 ⊆ (𝐺𝐴))
14451ffund 6672 . . . . . . . . . 10 (𝜑 → Fun 𝐺)
145143, 48sstrdi 3956 . . . . . . . . . 10 (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146 funimass3 7004 . . . . . . . . . 10 ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
147144, 145, 146syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (𝐺𝐴)))
148143, 147mpbird 256 . . . . . . . 8 (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149133, 148eqsstrd 3982 . . . . . . 7 (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
15021, 49cnf 22597 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
15177, 150syl 17 . . . . . . . . 9 (𝜑𝐹:𝐵 𝐽)
152151ffund 6672 . . . . . . . 8 (𝜑 → Fun 𝐹)
15329, 21cnf 22597 . . . . . . . . . . 11 (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
15472, 153syl 17 . . . . . . . . . 10 (𝜑𝐼:(0[,]1)⟶𝐵)
155154frnd 6676 . . . . . . . . 9 (𝜑 → ran 𝐼𝐵)
156151fdmd 6679 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐵)
157155, 156sseqtrrd 3985 . . . . . . . 8 (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158 funimass3 7004 . . . . . . . 8 ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
159152, 157, 158syl2anc 584 . . . . . . 7 (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (𝐹𝐴)))
160149, 159mpbid 231 . . . . . 6 (𝜑 → ran 𝐼 ⊆ (𝐹𝐴))
161 cnvimass 6033 . . . . . . 7 (𝐹𝐴) ⊆ dom 𝐹
162161, 151fssdm 6688 . . . . . 6 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
163 cnrest2 22637 . . . . . 6 ((𝐶 ∈ (TopOn‘𝐵) ∧ ran 𝐼 ⊆ (𝐹𝐴) ∧ (𝐹𝐴) ⊆ 𝐵) → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶t (𝐹𝐴)))))
164129, 160, 162, 163syl3anc 1371 . . . . 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 33861 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇𝐶)
169166, 168syl 17 . . . . . 6 (𝜑𝑇𝐶)
170 cvmlift3lem7.1 . . . . . . . . 9 (𝜑 → (𝐺𝑋) ∈ 𝐴)
17166, 170eqeltrd 2838 . . . . . . . 8 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
172 cvmlift3lem7.w . . . . . . . . 9 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
173167, 21, 172cvmsiota 33871 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
17423, 166, 56, 171, 173syl13anc 1372 . . . . . . 7 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
175174simpld 495 . . . . . 6 (𝜑𝑊𝑇)
176169, 175sseldd 3945 . . . . 5 (𝜑𝑊𝐶)
177 elssuni 4898 . . . . . . 7 (𝑊𝑇𝑊 𝑇)
178175, 177syl 17 . . . . . 6 (𝜑𝑊 𝑇)
179167cvmsuni 33863 . . . . . . 7 (𝑇 ∈ (𝑆𝐴) → 𝑇 = (𝐹𝐴))
180166, 179syl 17 . . . . . 6 (𝜑 𝑇 = (𝐹𝐴))
181178, 180sseqtrd 3984 . . . . 5 (𝜑𝑊 ⊆ (𝐹𝐴))
182167cvmsrcl 33858 . . . . . . . 8 (𝑇 ∈ (𝑆𝐴) → 𝐴𝐽)
183166, 182syl 17 . . . . . . 7 (𝜑𝐴𝐽)
184 cnima 22616 . . . . . . 7 ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐶)
18577, 183, 184syl2anc 584 . . . . . 6 (𝜑 → (𝐹𝐴) ∈ 𝐶)
186 restopn2 22528 . . . . . 6 ((𝐶 ∈ Top ∧ (𝐹𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
187127, 185, 186syl2anc 584 . . . . 5 (𝜑 → (𝑊 ∈ (𝐶t (𝐹𝐴)) ↔ (𝑊𝐶𝑊 ⊆ (𝐹𝐴))))
188176, 181, 187mpbir2and 711 . . . 4 (𝜑𝑊 ∈ (𝐶t (𝐹𝐴)))
189167cvmscld 33867 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝐴) ∧ 𝑊𝑇) → 𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19023, 166, 175, 189syl3anc 1371 . . . 4 (𝜑𝑊 ∈ (Clsd‘(𝐶t (𝐹𝐴))))
19133a1i 11 . . . 4 (𝜑 → 0 ∈ (0[,]1))
192174simprd 496 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝑊)
19374, 192eqeltrd 2838 . . . 4 (𝜑 → (𝐼‘0) ∈ 𝑊)
19429, 125, 165, 188, 190, 191, 193conncn 22777 . . 3 (𝜑𝐼:(0[,]1)⟶𝑊)
195 1elunit 13387 . . 3 1 ∈ (0[,]1)
196 ffvelcdm 7032 . . 3 ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197194, 195, 196sylancl 586 . 2 (𝜑 → (𝐼‘1) ∈ 𝑊)
198123, 197eqeltrd 2838 1 (𝜑 → (𝐻𝑍) ∈ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351  {crab 3407  cdif 3907  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  0cc0 11051  1c1 11052  [,]cicc 13267  t crest 17302  Topctop 22242  TopOnctopon 22259  Clsdccld 22367   Cn ccn 22575  Conncconn 22762  𝑛-Locally cnlly 22816  Homeochmeo 23104  IIcii 24238  *𝑝cpco 24363  PConncpconn 33813  SConncsconn 33814   CovMap ccvm 33849
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-ec 8650  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-cn 22578  df-cnp 22579  df-cmp 22738  df-conn 22763  df-lly 22817  df-nlly 22818  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-ii 24240  df-htpy 24333  df-phtpy 24334  df-phtpc 24355  df-pco 24368  df-pconn 33815  df-sconn 33816  df-cvm 33850
This theorem is referenced by:  cvmlift3lem7  33919
  Copyright terms: Public domain W3C validator