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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem1 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 33161. In cvmliftlem15 33160, we picked an 𝑁 large enough so that the sections (𝐺 “ [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁]) are all contained in an even covering, and the function 𝑇 enumerates these even coverings. So 1st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁]), and 2nd ‘(𝑇‘𝑀) is an even covering of 1st ‘(𝑇‘𝑀), which is to say a disjoint union of open sets in 𝐶 whose image is 1st ‘(𝑇‘𝑀). (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
cvmliftlem.1 | ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) |
cvmliftlem.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmliftlem.x | ⊢ 𝑋 = ∪ 𝐽 |
cvmliftlem.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmliftlem.g | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
cvmliftlem.p | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
cvmliftlem.e | ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) |
cvmliftlem.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
cvmliftlem.t | ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
cvmliftlem.a | ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) |
cvmliftlem.l | ⊢ 𝐿 = (topGen‘ran (,)) |
cvmliftlem1.m | ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) |
Ref | Expression |
---|---|
cvmliftlem1 | ⊢ ((𝜑 ∧ 𝜓) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5598 | . . . . . 6 ⊢ Rel ({𝑗} × (𝑆‘𝑗)) | |
2 | 1 | rgenw 3075 | . . . . 5 ⊢ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗)) |
3 | reliun 5715 | . . . . 5 ⊢ (Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗))) | |
4 | 2, 3 | mpbir 230 | . . . 4 ⊢ Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) |
5 | cvmliftlem.t | . . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
7 | cvmliftlem1.m | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
8 | 6, 7 | ffvelrnd 6944 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
9 | 1st2nd 7853 | . . . 4 ⊢ ((Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) → (𝑇‘𝑀) = 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉) | |
10 | 4, 8, 9 | sylancr 586 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) = 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉) |
11 | 10, 8 | eqeltrrd 2840 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
12 | fveq2 6756 | . . . 4 ⊢ (𝑗 = (1st ‘(𝑇‘𝑀)) → (𝑆‘𝑗) = (𝑆‘(1st ‘(𝑇‘𝑀)))) | |
13 | 12 | opeliunxp2 5736 | . . 3 ⊢ (〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ((1st ‘(𝑇‘𝑀)) ∈ 𝐽 ∧ (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))) |
14 | 13 | simprbi 496 | . 2 ⊢ (〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀)))) |
15 | 11, 14 | syl 17 | 1 ⊢ ((𝜑 ∧ 𝜓) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 {csn 4558 〈cop 4564 ∪ cuni 4836 ∪ ciun 4921 ↦ cmpt 5153 × cxp 5578 ◡ccnv 5579 ran crn 5581 ↾ cres 5582 “ cima 5583 Rel wrel 5585 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1st c1st 7802 2nd c2nd 7803 0cc0 10802 1c1 10803 − cmin 11135 / cdiv 11562 ℕcn 11903 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ↾t crest 17048 topGenctg 17065 Cn ccn 22283 Homeochmeo 22812 IIcii 23944 CovMap ccvm 33117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-1st 7804 df-2nd 7805 |
This theorem is referenced by: cvmliftlem6 33152 cvmliftlem8 33154 cvmliftlem9 33155 |
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