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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem3 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 31281. 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 481 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) |
4 | oveq1 6657 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 − 1) = (𝑀 − 1)) | |
5 | 4 | oveq1d 6665 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝑘 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁)) |
6 | oveq1 6657 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝑘 / 𝑁) = (𝑀 / 𝑁)) | |
7 | 5, 6 | oveq12d 6668 | . . . . . . 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 2674 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁)) = 𝑊) |
10 | 9 | imaeq2d 5466 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) = (𝐺 “ 𝑊)) |
11 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (𝑇‘𝑘) = (𝑇‘𝑀)) | |
12 | 11 | fveq2d 6195 | . . . . 5 ⊢ (𝑘 = 𝑀 → (1st ‘(𝑇‘𝑘)) = (1st ‘(𝑇‘𝑀))) |
13 | 10, 12 | sseq12d 3634 | . . . 4 ⊢ (𝑘 = 𝑀 → ((𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)) ↔ (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀)))) |
14 | 13 | rspcv 3305 | . . 3 ⊢ (𝑀 ∈ (1...𝑁) → (∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)) → (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀)))) |
15 | 1, 3, 14 | sylc 65 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀))) |
16 | cvmliftlem3.m | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝑊) | |
17 | cvmliftlem.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
18 | iiuni 22684 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
19 | cvmliftlem.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
20 | 18, 19 | cnf 21050 | . . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
21 | 17, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋) |
22 | 21 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋) |
23 | ffun 6048 | . . . . 5 ⊢ (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺) | |
24 | 22, 23 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → Fun 𝐺) |
25 | 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))))}) | |
26 | cvmliftlem.b | . . . . . 6 ⊢ 𝐵 = ∪ 𝐶 | |
27 | cvmliftlem.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
28 | cvmliftlem.p | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
29 | cvmliftlem.e | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) | |
30 | cvmliftlem.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
31 | cvmliftlem.t | . . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) | |
32 | cvmliftlem.l | . . . . . 6 ⊢ 𝐿 = (topGen‘ran (,)) | |
33 | 25, 26, 19, 27, 17, 28, 29, 30, 31, 2, 32, 1, 8 | cvmliftlem2 31268 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
34 | fdm 6051 | . . . . . 6 ⊢ (𝐺:(0[,]1)⟶𝑋 → dom 𝐺 = (0[,]1)) | |
35 | 22, 34 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → dom 𝐺 = (0[,]1)) |
36 | 33, 35 | sseqtr4d 3642 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ dom 𝐺) |
37 | funfvima2 6493 | . . . 4 ⊢ ((Fun 𝐺 ∧ 𝑊 ⊆ dom 𝐺) → (𝐴 ∈ 𝑊 → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊))) | |
38 | 24, 36, 37 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∈ 𝑊 → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊))) |
39 | 16, 38 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊)) |
40 | 15, 39 | sseldd 3604 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 ∪ ciun 4520 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 0cc0 9936 1c1 9937 − cmin 10266 / cdiv 10684 ℕcn 11020 (,)cioo 12175 [,]cicc 12178 ...cfz 12326 ↾t crest 16081 topGenctg 16098 Cn ccn 21028 Homeochmeo 21556 IIcii 22678 CovMap ccvm 31237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-ii 22680 |
This theorem is referenced by: cvmliftlem6 31272 cvmliftlem8 31274 cvmliftlem9 31275 |
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