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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for cvmlift 35304. In cvmliftlem15 35303, 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 5703 | . . . . . 6 ⊢ Rel ({𝑗} × (𝑆‘𝑗)) | |
| 2 | 1 | rgenw 3065 | . . . . 5 ⊢ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗)) |
| 3 | reliun 5826 | . . . . 5 ⊢ (Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗))) | |
| 4 | 2, 3 | mpbir 231 | . . . 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 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
| 9 | 1st2nd 8064 | . . . 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 587 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) = 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉) |
| 11 | 10, 8 | eqeltrrd 2842 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 〈(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))〉 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
| 12 | fveq2 6906 | . . . 4 ⊢ (𝑗 = (1st ‘(𝑇‘𝑀)) → (𝑆‘𝑗) = (𝑆‘(1st ‘(𝑇‘𝑀)))) | |
| 13 | 12 | opeliunxp2 5849 | . . 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 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 〈cop 4632 ∪ cuni 4907 ∪ ciun 4991 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 0cc0 11155 1c1 11156 − cmin 11492 / cdiv 11920 ℕcn 12266 (,)cioo 13387 [,]cicc 13390 ...cfz 13547 ↾t crest 17465 topGenctg 17482 Cn ccn 23232 Homeochmeo 23761 IIcii 24901 CovMap ccvm 35260 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: cvmliftlem6 35295 cvmliftlem8 35297 cvmliftlem9 35298 |
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