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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem1 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 31828. In cvmliftlem15 31827, 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 5361 | . . . . . 6 ⊢ Rel ({𝑗} × (𝑆‘𝑗)) | |
2 | 1 | rgenw 3134 | . . . . 5 ⊢ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗)) |
3 | reliun 5475 | . . . . 5 ⊢ (Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗))) | |
4 | 2, 3 | mpbir 223 | . . . 4 ⊢ Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) |
5 | cvmliftlem.t | . . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) | |
6 | 5 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
7 | cvmliftlem1.m | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
8 | 6, 7 | ffvelrnd 6610 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
9 | 1st2nd 7477 | . . . 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 583 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) = 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉) |
11 | 10, 8 | eqeltrrd 2908 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
12 | fveq2 6434 | . . . 4 ⊢ (𝑗 = (1st ‘(𝑇‘𝑀)) → (𝑆‘𝑗) = (𝑆‘(1st ‘(𝑇‘𝑀)))) | |
13 | 12 | opeliunxp2 5494 | . . 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 492 | . 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 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 {crab 3122 ∖ cdif 3796 ∩ cin 3798 ⊆ wss 3799 ∅c0 4145 𝒫 cpw 4379 {csn 4398 〈cop 4404 ∪ cuni 4659 ∪ ciun 4741 ↦ cmpt 4953 × cxp 5341 ◡ccnv 5342 ran crn 5344 ↾ cres 5345 “ cima 5346 Rel wrel 5348 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 1st c1st 7427 2nd c2nd 7428 0cc0 10253 1c1 10254 − cmin 10586 / cdiv 11010 ℕcn 11351 (,)cioo 12464 [,]cicc 12467 ...cfz 12620 ↾t crest 16435 topGenctg 16452 Cn ccn 21400 Homeochmeo 21928 IIcii 23049 CovMap ccvm 31784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-fv 6132 df-1st 7429 df-2nd 7430 |
This theorem is referenced by: cvmliftlem6 31819 cvmliftlem8 31821 cvmliftlem9 31822 |
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