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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem3 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 30535. 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) |
4 | oveq1 6556 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 − 1) = (𝑀 − 1)) | |
5 | 4 | oveq1d 6564 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝑘 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁)) |
6 | oveq1 6556 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝑘 / 𝑁) = (𝑀 / 𝑁)) | |
7 | 5, 6 | oveq12d 6567 | . . . . . . 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 2662 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁)) = 𝑊) |
10 | 9 | imaeq2d 5385 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) = (𝐺 “ 𝑊)) |
11 | fveq2 6103 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (𝑇‘𝑘) = (𝑇‘𝑀)) | |
12 | 11 | fveq2d 6107 | . . . . 5 ⊢ (𝑘 = 𝑀 → (1st ‘(𝑇‘𝑘)) = (1st ‘(𝑇‘𝑀))) |
13 | 10, 12 | sseq12d 3597 | . . . 4 ⊢ (𝑘 = 𝑀 → ((𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)) ↔ (𝐺 “ 𝑊) ⊆ (1st ‘(𝑇‘𝑀)))) |
14 | 13 | rspcv 3278 | . . 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 63 | . 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 22492 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
19 | cvmliftlem.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
20 | 18, 19 | cnf 20860 | . . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
21 | 17, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋) |
22 | 21 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋) |
23 | ffun 5961 | . . . . 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 30522 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
34 | fdm 5964 | . . . . . 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 3605 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ dom 𝐺) |
37 | funfvima2 6397 | . . . 4 ⊢ ((Fun 𝐺 ∧ 𝑊 ⊆ dom 𝐺) → (𝐴 ∈ 𝑊 → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊))) | |
38 | 24, 36, 37 | syl2anc 691 | . . 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 3569 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ∪ ciun 4455 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 0cc0 9815 1c1 9816 − cmin 10145 / cdiv 10563 ℕcn 10897 (,)cioo 12046 [,]cicc 12049 ...cfz 12197 ↾t crest 15904 topGenctg 15921 Cn ccn 20838 Homeochmeo 21366 IIcii 22486 CovMap ccvm 30491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-ii 22488 |
This theorem is referenced by: cvmliftlem6 30526 cvmliftlem8 30528 cvmliftlem9 30529 |
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