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Theorem cvmlift3lem7 33919
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)
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 33915 . . . 4 (𝜑𝐻:𝑌𝐵)
131, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem5 33917 . . . . 5 (𝜑 → (𝐹𝐻) = 𝐺)
1413, 8eqeltrd 2838 . . . 4 (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))
15 sconntop 33822 . . . . 5 (𝐾 ∈ SConn → 𝐾 ∈ Top)
165, 15syl 17 . . . 4 (𝜑𝐾 ∈ Top)
17 cvmlift3lem7.3 . . . . . 6 (𝜑𝑀 ⊆ (𝐺𝐴))
18 cnvimass 6033 . . . . . . 7 (𝐺𝐴) ⊆ dom 𝐺
19 eqid 2736 . . . . . . . . 9 𝐽 = 𝐽
202, 19cnf 22597 . . . . . . . 8 (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌 𝐽)
21 fdm 6677 . . . . . . . 8 (𝐺:𝑌 𝐽 → dom 𝐺 = 𝑌)
228, 20, 213syl 18 . . . . . . 7 (𝜑 → dom 𝐺 = 𝑌)
2318, 22sseqtrid 3996 . . . . . 6 (𝜑 → (𝐺𝐴) ⊆ 𝑌)
2417, 23sstrd 3954 . . . . 5 (𝜑𝑀𝑌)
25 cvmlift3lem7.5 . . . . . 6 (𝜑𝑉𝑀)
26 cvmlift3lem7.6 . . . . . 6 (𝜑𝑋𝑉)
2725, 26sseldd 3945 . . . . 5 (𝜑𝑋𝑀)
2824, 27sseldd 3945 . . . 4 (𝜑𝑋𝑌)
29 cvmlift3lem7.2 . . . 4 (𝜑𝑇 ∈ (𝑆𝐴))
3012, 28ffvelcdmd 7036 . . . . 5 (𝜑 → (𝐻𝑋) ∈ 𝐵)
31 fvco3 6940 . . . . . . . 8 ((𝐻:𝑌𝐵𝑋𝑌) → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3212, 28, 31syl2anc 584 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐹‘(𝐻𝑋)))
3313fveq1d 6844 . . . . . . 7 (𝜑 → ((𝐹𝐻)‘𝑋) = (𝐺𝑋))
3432, 33eqtr3d 2778 . . . . . 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 33871 . . . . 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 2736 . . . . . . . . . . 11 (𝐻𝑋) = (𝐻𝑋)
411, 2, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem4 33916 . . . . . . . . . . 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 6841 . . . . . . . . . . 11 (𝑓 = → (𝑓‘0) = (‘0))
4645eqeq1d 2738 . . . . . . . . . 10 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
47 fveq1 6841 . . . . . . . . . . 11 (𝑓 = → (𝑓‘1) = (‘1))
4847eqeq1d 2738 . . . . . . . . . 10 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
49 coeq2 5814 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝐺𝑓) = (𝐺))
5049eqeq2d 2747 . . . . . . . . . . . . . . 15 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
5150anbi1d 630 . . . . . . . . . . . . . 14 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
5251riotabidv 7315 . . . . . . . . . . . . 13 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
53 coeq2 5814 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝐹𝑎) = (𝐹𝑔))
5453eqeq1d 2738 . . . . . . . . . . . . . . 15 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺) ↔ (𝐹𝑔) = (𝐺)))
55 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑔 → (𝑎‘0) = (𝑔‘0))
5655eqeq1d 2738 . . . . . . . . . . . . . . 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 7323 . . . . . . . . . . . . 13 (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))
5952, 58eqtr4di 2794 . . . . . . . . . . . 12 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃)))
6059fveq1d 6844 . . . . . . . . . . 11 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑎 ∈ (II Cn 𝐶)((𝐹𝑎) = (𝐺) ∧ (𝑎‘0) = 𝑃))‘1))
6160eqeq1d 2738 . . . . . . . . . 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 3226 . . . . . . . 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 22513 . . . . . . . . . . 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 2736 . . . . . . . . 9 (𝐾t 𝑀) = (𝐾t 𝑀)
7473pconncn 33818 . . . . . . . 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 3217 . . . . . . . 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 2738 . . . . . . . . . . . . 13 (𝑎 = 𝑔 → ((𝐹𝑎) = (𝐺𝑛) ↔ (𝐹𝑔) = (𝐺𝑛)))
9455eqeq1d 2738 . . . . . . . . . . . . 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 7323 . . . . . . . . . . 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 33918 . . . . . . . . . 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 3205 . . . . . . . 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 3143 . . . . 5 (𝜑 → ∀𝑦𝑀 (𝐻𝑦) ∈ 𝑊)
10312ffund 6672 . . . . . 6 (𝜑 → Fun 𝐻)
10412fdmd 6679 . . . . . . 7 (𝜑 → dom 𝐻 = 𝑌)
10524, 104sseqtrrd 3985 . . . . . 6 (𝜑𝑀 ⊆ dom 𝐻)
106 funimass4 6907 . . . . . 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 33897 . . 3 (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))
11073cncnpi 22629 . . 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 22409 . . . . 5 (((𝐾 ∈ Top ∧ 𝑀𝑌) ∧ (𝑉𝐾𝑉𝑀)) → 𝑉 ⊆ ((int‘𝐾)‘𝑀))
11416, 24, 112, 25, 113syl22anc 837 . . . 4 (𝜑𝑉 ⊆ ((int‘𝐾)‘𝑀))
115114, 26sseldd 3945 . . 3 (𝜑𝑋 ∈ ((int‘𝐾)‘𝑀))
1162, 1cnprest 22640 . . 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 3064  wrex 3073  {crab 3407  cdif 3907  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865  cmpt 5188  ccnv 5632  dom cdm 5633  cres 5635  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  0cc0 11051  1c1 11052  t crest 17302  Topctop 22242  intcnt 22368   Cn ccn 22575   CnP ccnp 22576  𝑛-Locally cnlly 22816  Homeochmeo 23104  IIcii 24238  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:  cvmlift3lem8  33920
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