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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem3 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 32548. 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 7165 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 − 1) = (𝑀 − 1)) | |
5 | 4 | oveq1d 7173 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝑘 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁)) |
6 | oveq1 7165 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝑘 / 𝑁) = (𝑀 / 𝑁)) | |
7 | 5, 6 | oveq12d 7176 | . . . . . . 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 2876 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁)) = 𝑊) |
10 | 9 | imaeq2d 5931 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) = (𝐺 “ 𝑊)) |
11 | 2fveq3 6677 | . . . . 5 ⊢ (𝑘 = 𝑀 → (1st ‘(𝑇‘𝑘)) = (1st ‘(𝑇‘𝑀))) | |
12 | 10, 11 | sseq12d 4002 | . . . 4 ⊢ (𝑘 = 𝑀 → ((𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)) ↔ (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀)))) |
13 | 12 | rspcv 3620 | . . 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 23491 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
18 | cvmliftlem.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
19 | 17, 18 | cnf 21856 | . . . . . . 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 6520 | . . . 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 32535 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
32 | 21 | fdmd 6525 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → dom 𝐺 = (0[,]1)) |
33 | 31, 32 | sseqtrrd 4010 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ dom 𝐺) |
34 | funfvima2 6995 | . . . 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 3970 | 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 3140 {crab 3144 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 ∪ cuni 4840 ∪ ciun 4921 ↦ cmpt 5148 × cxp 5555 ◡ccnv 5556 dom cdm 5557 ran crn 5558 ↾ cres 5559 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 0cc0 10539 1c1 10540 − cmin 10872 / cdiv 11299 ℕcn 11640 (,)cioo 12741 [,]cicc 12744 ...cfz 12895 ↾t crest 16696 topGenctg 16713 Cn ccn 21834 Homeochmeo 22363 IIcii 23485 CovMap ccvm 32504 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-cn 21837 df-ii 23487 |
This theorem is referenced by: cvmliftlem6 32539 cvmliftlem8 32541 cvmliftlem9 32542 |
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