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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem3 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 32546. Since 1st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ 𝑊), every element 𝐴 ∈ 𝑊 satisfies (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀)). (Contributed by Mario Carneiro, 16-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)) |
cvmliftlem3.3 | ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) |
cvmliftlem3.m | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝑊) |
Ref | Expression |
---|---|
cvmliftlem3 | ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftlem1.m | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
2 | cvmliftlem.a | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) |
4 | oveq1 7163 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 − 1) = (𝑀 − 1)) | |
5 | 4 | oveq1d 7171 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝑘 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁)) |
6 | oveq1 7163 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝑘 / 𝑁) = (𝑀 / 𝑁)) | |
7 | 5, 6 | oveq12d 7174 | . . . . . . 7 ⊢ (𝑘 = 𝑀 → (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁)) = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))) |
8 | cvmliftlem3.3 | . . . . . . 7 ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) | |
9 | 7, 8 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁)) = 𝑊) |
10 | 9 | imaeq2d 5929 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) = (𝐺 “ 𝑊)) |
11 | 2fveq3 6675 | . . . . 5 ⊢ (𝑘 = 𝑀 → (1st ‘(𝑇‘𝑘)) = (1st ‘(𝑇‘𝑀))) | |
12 | 10, 11 | sseq12d 4000 | . . . 4 ⊢ (𝑘 = 𝑀 → ((𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)) ↔ (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀)))) |
13 | 12 | rspcv 3618 | . . 3 ⊢ (𝑀 ∈ (1...𝑁) → (∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)) → (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀)))) |
14 | 1, 3, 13 | sylc 65 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀))) |
15 | cvmliftlem3.m | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝑊) | |
16 | cvmliftlem.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
17 | iiuni 23489 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
18 | cvmliftlem.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
19 | 17, 18 | cnf 21854 | . . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
20 | 16, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋) |
21 | 20 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋) |
22 | 21 | ffund 6518 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → Fun 𝐺) |
23 | cvmliftlem.1 | . . . . . 6 ⊢ 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))))}) | |
24 | cvmliftlem.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐶 | |
25 | cvmliftlem.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
26 | cvmliftlem.p | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
27 | cvmliftlem.e | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) | |
28 | cvmliftlem.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
29 | cvmliftlem.t | . . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) | |
30 | cvmliftlem.l | . . . . . 6 ⊢ 𝐿 = (topGen‘ran (,)) | |
31 | 23, 24, 18, 25, 16, 26, 27, 28, 29, 2, 30, 1, 8 | cvmliftlem2 32533 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
32 | 21 | fdmd 6523 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → dom 𝐺 = (0[,]1)) |
33 | 31, 32 | sseqtrrd 4008 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ dom 𝐺) |
34 | funfvima2 6993 | . . . 4 ⊢ ((Fun 𝐺 ∧ 𝑊 ⊆ dom 𝐺) → (𝐴 ∈ 𝑊 → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊))) | |
35 | 22, 33, 34 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∈ 𝑊 → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊))) |
36 | 15, 35 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊)) |
37 | 14, 36 | sseldd 3968 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ∪ ciun 4919 ↦ cmpt 5146 × cxp 5553 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 “ cima 5558 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 0cc0 10537 1c1 10538 − cmin 10870 / cdiv 11297 ℕcn 11638 (,)cioo 12739 [,]cicc 12742 ...cfz 12893 ↾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 ax-pre-sup 10615 |
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-rmo 3146 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-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-bases 21554 df-cn 21835 df-ii 23485 |
This theorem is referenced by: cvmliftlem6 32537 cvmliftlem8 32539 cvmliftlem9 32540 |
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