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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem4 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 32546. The function 𝑄 will be our lifted path, defined piecewise on each section [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] for 𝑀 ∈ (1...𝑁). For 𝑀 = 0, it is a "seed" value which makes the rest of the recursion work, a singleton function mapping 0 to 𝑃. (Contributed by Mario Carneiro, 15-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 (,)) |
cvmliftlem.q | ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})) |
Ref | Expression |
---|---|
cvmliftlem4 | ⊢ (𝑄‘0) = {〈0, 𝑃〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftlem.q | . . . . 5 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})) | |
2 | 1 | fveq1i 6671 | . . . 4 ⊢ (𝑄‘0) = (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉}))‘0) |
3 | 0z 11993 | . . . . 5 ⊢ 0 ∈ ℤ | |
4 | seq1 13383 | . . . . 5 ⊢ (0 ∈ ℤ → (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉}))‘0) = ((( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})‘0)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉}))‘0) = ((( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})‘0) |
6 | 2, 5 | eqtri 2844 | . . 3 ⊢ (𝑄‘0) = ((( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})‘0) |
7 | fnresi 6476 | . . . 4 ⊢ ( I ↾ ℕ) Fn ℕ | |
8 | c0ex 10635 | . . . . 5 ⊢ 0 ∈ V | |
9 | snex 5332 | . . . . 5 ⊢ {〈0, 𝑃〉} ∈ V | |
10 | 8, 9 | fnsn 6412 | . . . 4 ⊢ {〈0, {〈0, 𝑃〉}〉} Fn {0} |
11 | 0nnn 11674 | . . . . . 6 ⊢ ¬ 0 ∈ ℕ | |
12 | disjsn 4647 | . . . . . 6 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) | |
13 | 11, 12 | mpbir 233 | . . . . 5 ⊢ (ℕ ∩ {0}) = ∅ |
14 | 8 | snid 4601 | . . . . 5 ⊢ 0 ∈ {0} |
15 | 13, 14 | pm3.2i 473 | . . . 4 ⊢ ((ℕ ∩ {0}) = ∅ ∧ 0 ∈ {0}) |
16 | fvun2 6755 | . . . 4 ⊢ ((( I ↾ ℕ) Fn ℕ ∧ {〈0, {〈0, 𝑃〉}〉} Fn {0} ∧ ((ℕ ∩ {0}) = ∅ ∧ 0 ∈ {0})) → ((( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})‘0) = ({〈0, {〈0, 𝑃〉}〉}‘0)) | |
17 | 7, 10, 15, 16 | mp3an 1457 | . . 3 ⊢ ((( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})‘0) = ({〈0, {〈0, 𝑃〉}〉}‘0) |
18 | 6, 17 | eqtri 2844 | . 2 ⊢ (𝑄‘0) = ({〈0, {〈0, 𝑃〉}〉}‘0) |
19 | 8, 9 | fvsn 6943 | . 2 ⊢ ({〈0, {〈0, 𝑃〉}〉}‘0) = {〈0, 𝑃〉} |
20 | 18, 19 | eqtri 2844 | 1 ⊢ (𝑄‘0) = {〈0, 𝑃〉} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 〈cop 4573 ∪ cuni 4838 ∪ ciun 4919 ↦ cmpt 5146 I cid 5459 × cxp 5553 ◡ccnv 5554 ran crn 5556 ↾ cres 5557 “ cima 5558 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7687 2nd c2nd 7688 0cc0 10537 1c1 10538 − cmin 10870 / cdiv 11297 ℕcn 11638 ℤcz 11982 (,)cioo 12739 [,]cicc 12742 ...cfz 12893 seqcseq 13370 ↾t crest 16694 topGenctg 16711 Cn ccn 21832 Homeochmeo 22361 IIcii 23483 CovMap ccvm 32502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 |
This theorem is referenced by: cvmliftlem7 32538 cvmliftlem13 32543 |
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