Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3lem7 Structured version   Visualization version   GIF version

Theorem cvmlift3lem7 33731
Description: Lemma for cvmlift3 33734. (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)
cvmlift3lem7.7 (𝜑 → (𝐾t 𝑀) ∈ PConn)
cvmlift3lem7.4 (𝜑𝑉𝐾)
cvmlift3lem7.5 (𝜑𝑉𝑀)
cvmlift3lem7.6 (𝜑𝑋𝑉)
Assertion
Ref Expression
cvmlift3lem7 (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Distinct variable groups:   𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝐴   𝑓,𝑔,𝑧,𝑏,𝑥   𝐽,𝑏   𝑔,𝑐,𝑥,𝐽,𝑑,𝑓,𝑘,𝑠   𝐹,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠   𝑥,𝑧,𝐹   𝑓,𝑀,𝑔,𝑥   𝐻,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑆,𝑏,𝑓,𝑥   𝐵,𝑏,𝑑,𝑓,𝑔,𝑥,𝑧   𝑋,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝐺,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧   𝑇,𝑏,𝑐,𝑑,𝑠   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)   𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝑇(𝑥,𝑧,𝑓,𝑔,𝑘)   𝐺(𝑠)   𝐻(𝑘,𝑠)   𝐽(𝑧)   𝐾(𝑘,𝑠,𝑑)   𝑀(𝑧,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑂(𝑘,𝑠,𝑑)   𝑉(𝑥,𝑧,𝑓,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑)   𝑊(𝑧,𝑔,𝑘,𝑠,𝑏)   𝑋(𝑘,𝑠)   𝑌(𝑘,𝑠,𝑏,𝑐,𝑑)

Proof of Theorem cvmlift3lem7
Dummy variables 𝑎 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift3.b . . . 4 𝐵 = 𝐶
2 cvmlift3.y . . . 4 𝑌 = 𝐾
3 cvmlift3lem7.s . . . 4 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})
4 cvmlift3.f . . . 4 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
5 cvmlift3.k . . . . 5 (𝜑𝐾 ∈ SConn)
6 cvmlift3.l . . . . 5 (𝜑𝐾 ∈ 𝑛-Locally PConn)
7 cvmlift3.o . . . . 5 (𝜑𝑂𝑌)
8 cvmlift3.g . . . . 5 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
9 cvmlift3.p . . . . 5 (𝜑𝑃𝐵)
10 cvmlift3.e . . . . 5 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
11 cvmlift3.h . . . . 5 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
121, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem3 33727 . . . 4 (𝜑𝐻:𝑌𝐵)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 33729 . . . . 5 (𝜑 → (𝐹𝐻) = 𝐺)
1413, 8eqeltrd 2838 . . . 4 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 33634 . . . . 5 (𝐾 ∈ SConn → 𝐾 ∈ Top)
165, 15syl 17 . . . 4 (𝜑𝐾 ∈ Top)
17 cvmlift3lem7.3 . . . . . 6 (𝜑𝑀 ⊆ (𝐺𝐴))
18 cnvimass 6031 . . . . . . 7 (𝐺𝐴) ⊆ dom 𝐺
19 eqid 2737 . . . . . . . . 9 𝐽 = 𝐽
202, 19cnf 22549 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
21 fdm 6674 . . . . . . . 8 (𝐺:𝑌 𝐽 → dom 𝐺 = 𝑌)
228, 20, 213syl 18 . . . . . . 7 (𝜑 → dom 𝐺 = 𝑌)
2318, 22sseqtrid 3994 . . . . . 6 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
2417, 23sstrd 3952 . . . . 5 (𝜑𝑀𝑌)
25 cvmlift3lem7.5 . . . . . 6 (𝜑𝑉𝑀)
26 cvmlift3lem7.6 . . . . . 6 (𝜑𝑋𝑉)
2725, 26sseldd 3943 . . . . 5 (𝜑𝑋𝑀)
2824, 27sseldd 3943 . . . 4 (𝜑𝑋𝑌)
29 cvmlift3lem7.2 . . . 4 (𝜑𝑇 ∈ (𝑆𝐴))
3012, 28ffvelcdmd 7032 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝐵)
31 fvco3 6937 . . . . . . . 8 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3212, 28, 31syl2anc 584 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3313fveq1d 6841 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
3432, 33eqtr3d 2779 . . . . . 6 (𝜑 → (𝐹‘(𝐻𝑋)) = (𝐺𝑋))
35 cvmlift3lem7.1 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ 𝐴)
3634, 35eqeltrd 2838 . . . . 5 (𝜑 → (𝐹‘(𝐻𝑋)) ∈ 𝐴)
37 cvmlift3lem7.w . . . . . 6 𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)
383, 1, 37cvmsiota 33683 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝐴) ∧ (𝐻𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻𝑋)) ∈ 𝐴)) → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
394, 29, 30, 36, 38syl13anc 1372 . . . 4 (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))
40 eqid 2737 . . . . . . . . . . 11 (𝐻𝑋) = (𝐻𝑋)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 33728 . . . . . . . . . . 11 ((𝜑𝑋𝑌) → ((𝐻𝑋) = (𝐻𝑋) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋))))
4240, 41mpbii 232 . . . . . . . . . 10 ((𝜑𝑋𝑌) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
4328, 42mpdan 685 . . . . . . . . 9 (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
4443adantr 481 . . . . . . . 8 ((𝜑𝑦𝑀) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)))
45 fveq1 6838 . . . . . . . . . . 11 (𝑓 = → (𝑓‘0) = (‘0))
4645eqeq1d 2739 . . . . . . . . . 10 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
47 fveq1 6838 . . . . . . . . . . 11 (𝑓 = → (𝑓‘1) = (‘1))
4847eqeq1d 2739 . . . . . . . . . 10 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
49 coeq2 5812 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝐺𝑓) = (𝐺))
5049eqeq2d 2748 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
5150anbi1d 630 . . . . . . . . . . . . . 14 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5251riotabidv 7309 . . . . . . . . . . . . 13 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
53 coeq2 5812 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝐹𝑎) = (𝐹𝑔))
5453eqeq1d 2739 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺) ↔ (𝐹𝑔) = (𝐺)))
55 fveq1 6838 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
5655eqeq1d 2739 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝑎‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
5754, 56anbi12d 631 . . . . . . . . . . . . . 14 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5857cbvriotavw 7317 . . . . . . . . . . . . 13 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))
5952, 58eqtr4di 2795 . . . . . . . . . . . 12 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)))
6059fveq1d 6841 . . . . . . . . . . 11 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1))
6160eqeq1d 2739 . . . . . . . . . 10 (𝑓 = → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋) ↔ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
6246, 48, 613anbi123d 1436 . . . . . . . . 9 (𝑓 = → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)) ↔ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋))))
6362cbvrexvw 3224 . . . . . . . 8 (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻𝑋)) ↔ ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
6444, 63sylib 217 . . . . . . 7 ((𝜑𝑦𝑀) → ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
65 cvmlift3lem7.7 . . . . . . . . 9 (𝜑 → (𝐾t 𝑀) ∈ PConn)
6665adantr 481 . . . . . . . 8 ((𝜑𝑦𝑀) → (𝐾t 𝑀) ∈ PConn)
672restuni 22465 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝑀𝑌) → 𝑀 = (𝐾t 𝑀))
6816, 24, 67syl2anc 584 . . . . . . . . . 10 (𝜑𝑀 = (𝐾t 𝑀))
6927, 68eleqtrd 2840 . . . . . . . . 9 (𝜑𝑋 (𝐾t 𝑀))
7069adantr 481 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑋 (𝐾t 𝑀))
7168eleq2d 2823 . . . . . . . . 9 (𝜑 → (𝑦𝑀𝑦 (𝐾t 𝑀)))
7271biimpa 477 . . . . . . . 8 ((𝜑𝑦𝑀) → 𝑦 (𝐾t 𝑀))
73 eqid 2737 . . . . . . . . 9 (𝐾t 𝑀) = (𝐾t 𝑀)
7473pconncn 33630 . . . . . . . 8 (((𝐾t 𝑀) ∈ PConn ∧ 𝑋 (𝐾t 𝑀) ∧ 𝑦 (𝐾t 𝑀)) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
7566, 70, 72, 74syl3anc 1371 . . . . . . 7 ((𝜑𝑦𝑀) → ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
76 reeanv 3215 . . . . . . . 8 (∃ ∈ (II Cn 𝐾)∃𝑛 ∈ (II Cn (𝐾t 𝑀))(((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) ↔ (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)))
774ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
785ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐾 ∈ SConn)
796ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐾 ∈ 𝑛-Locally PConn)
807ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑂𝑌)
818ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝐺 ∈ (𝐾 Cn 𝐽))
829ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑃𝐵)
8310ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐹𝑃) = (𝐺𝑂))
8435ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐺𝑋) ∈ 𝐴)
8529ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑇 ∈ (𝑆𝐴))
8617ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑀 ⊆ (𝐺𝐴))
8727ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑋𝑀)
88 simpllr 774 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑦𝑀)
89 simplrl 775 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ∈ (II Cn 𝐾))
90 simprl 769 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)))
91 simplrr 776 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → 𝑛 ∈ (II Cn (𝐾t 𝑀)))
92 simprr 771 . . . . . . . . . . 11 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))
9353eqeq1d 2739 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺𝑛) ↔ (𝐹𝑔) = (𝐺𝑛)))
9455eqeq1d 2739 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝑎‘0) = (𝐻𝑋) ↔ (𝑔‘0) = (𝐻𝑋)))
9593, 94anbi12d 631 . . . . . . . . . . . 12 (𝑎 = 𝑔 → (((𝐹𝑎) = (𝐺𝑛) ∧ (𝑎‘0) = (𝐻𝑋)) ↔ ((𝐹𝑔) = (𝐺𝑛) ∧ (𝑔‘0) = (𝐻𝑋))))
9695cbvriotavw 7317 . . . . . . . . . . 11 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺𝑛) ∧ (𝑎‘0) = (𝐻𝑋))) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑛) ∧ (𝑔‘0) = (𝐻𝑋)))
971, 2, 77, 78, 79, 80, 81, 82, 83, 11, 3, 84, 85, 86, 37, 87, 88, 89, 58, 90, 91, 92, 96cvmlift3lem6 33730 . . . . . . . . . 10 ((((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) ∧ (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦))) → (𝐻𝑦) ∈ 𝑊)
9897ex 413 . . . . . . . . 9 (((𝜑𝑦𝑀) ∧ ( ∈ (II Cn 𝐾) ∧ 𝑛 ∈ (II Cn (𝐾t 𝑀)))) → ((((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻𝑦) ∈ 𝑊))
9998rexlimdvva 3203 . . . . . . . 8 ((𝜑𝑦𝑀) → (∃ ∈ (II Cn 𝐾)∃𝑛 ∈ (II Cn (𝐾t 𝑀))(((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻𝑦) ∈ 𝑊))
10076, 99biimtrrid 242 . . . . . . 7 ((𝜑𝑦𝑀) → ((∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1) = (𝐻𝑋)) ∧ ∃𝑛 ∈ (II Cn (𝐾t 𝑀))((𝑛‘0) = 𝑋 ∧ (𝑛‘1) = 𝑦)) → (𝐻𝑦) ∈ 𝑊))
10164, 75, 100mp2and 697 . . . . . 6 ((𝜑𝑦𝑀) → (𝐻𝑦) ∈ 𝑊)
102101ralrimiva 3141 . . . . 5 (𝜑 → ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊)
10312ffund 6669 . . . . . 6 (𝜑 → Fun 𝐻)
10412fdmd 6676 . . . . . . 7 (𝜑 → dom 𝐻 = 𝑌)
10524, 104sseqtrrd 3983 . . . . . 6 (𝜑𝑀 ⊆ dom 𝐻)
106 funimass4 6904 . . . . . 6 ((Fun 𝐻𝑀 ⊆ dom 𝐻) → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
107103, 105, 106syl2anc 584 . . . . 5 (𝜑 → ((𝐻𝑀) ⊆ 𝑊 ↔ ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊))
108102, 107mpbird 256 . . . 4 (𝜑 → (𝐻𝑀) ⊆ 𝑊)
1091, 2, 3, 4, 12, 14, 16, 28, 29, 39, 24, 108cvmlift2lem9a 33709 . . 3 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
11073cncnpi 22581 . . 3 (((𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶) ∧ 𝑋 (𝐾t 𝑀)) → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
111109, 69, 110syl2anc 584 . 2 (𝜑 → (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋))
112 cvmlift3lem7.4 . . . . 5 (𝜑𝑉𝐾)
1132ssntr 22361 . . . . 5 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑉𝐾𝑉𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
11416, 24, 112, 25, 113syl22anc 837 . . . 4 (𝜑𝑉 ⊆ ((int‘𝐾)‘𝑀))
115114, 26sseldd 3943 . . 3 (𝜑𝑋 ∈ ((int‘𝐾)‘𝑀))
1162, 1cnprest 22592 . . 3 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑋 ∈ ((int‘𝐾)‘𝑀) ∧ 𝐻:𝑌𝐵)) → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
11716, 24, 115, 12, 116syl22anc 837 . 2 (𝜑 → (𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋) ↔ (𝐻𝑀) ∈ (((𝐾t 𝑀) CnP 𝐶)‘𝑋)))
118111, 117mpbird 256 1 (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wrex 3071  {crab 3405  cdif 3905  cin 3907  wss 3908  c0 4280  𝒫 cpw 4558  {csn 4584   cuni 4863  cmpt 5186  ccnv 5630  dom cdm 5631  cres 5633  cima 5634  ccom 5635  Fun wfun 6487  wf 6489  cfv 6493  crio 7306  (class class class)co 7351  0cc0 11009  1c1 11010  t crest 17262  Topctop 22194  intcnt 22320   Cn ccn 22527   CnP ccnp 22528  𝑛-Locally cnlly 22768  Homeochmeo 23056  IIcii 24190  PConncpconn 33625  SConncsconn 33626   CovMap ccvm 33661
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-inf2 9535  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088  ax-mulf 11089
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609  df-om 7795  df-1st 7913  df-2nd 7914  df-supp 8085  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-2o 8405  df-er 8606  df-ec 8608  df-map 8725  df-ixp 8794  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-fsupp 9264  df-fi 9305  df-sup 9336  df-inf 9337  df-oi 9404  df-card 9833  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-div 11771  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-q 12828  df-rp 12870  df-xneg 12987  df-xadd 12988  df-xmul 12989  df-ioo 13222  df-ico 13224  df-icc 13225  df-fz 13379  df-fzo 13522  df-fl 13651  df-seq 13861  df-exp 13922  df-hash 14185  df-cj 14944  df-re 14945  df-im 14946  df-sqrt 15080  df-abs 15081  df-clim 15330  df-sum 15531  df-struct 16979  df-sets 16996  df-slot 17014  df-ndx 17026  df-base 17044  df-ress 17073  df-plusg 17106  df-mulr 17107  df-starv 17108  df-sca 17109  df-vsca 17110  df-ip 17111  df-tset 17112  df-ple 17113  df-ds 17115  df-unif 17116  df-hom 17117  df-cco 17118  df-rest 17264  df-topn 17265  df-0g 17283  df-gsum 17284  df-topgen 17285  df-pt 17286  df-prds 17289  df-xrs 17344  df-qtop 17349  df-imas 17350  df-xps 17352  df-mre 17426  df-mrc 17427  df-acs 17429  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-submnd 18562  df-mulg 18832  df-cntz 19056  df-cmn 19523  df-psmet 20741  df-xmet 20742  df-met 20743  df-bl 20744  df-mopn 20745  df-cnfld 20750  df-top 22195  df-topon 22212  df-topsp 22234  df-bases 22248  df-cld 22322  df-ntr 22323  df-cls 22324  df-nei 22401  df-cn 22530  df-cnp 22531  df-cmp 22690  df-conn 22715  df-lly 22769  df-nlly 22770  df-tx 22865  df-hmeo 23058  df-xms 23625  df-ms 23626  df-tms 23627  df-ii 24192  df-htpy 24285  df-phtpy 24286  df-phtpc 24307  df-pco 24320  df-pconn 33627  df-sconn 33628  df-cvm 33662
This theorem is referenced by:  cvmlift3lem8  33732
  Copyright terms: Public domain W3C validator