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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem3 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 35284. 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 7438 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 − 1) = (𝑀 − 1)) | |
5 | 4 | oveq1d 7446 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝑘 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁)) |
6 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝑘 / 𝑁) = (𝑀 / 𝑁)) | |
7 | 5, 6 | oveq12d 7449 | . . . . . . 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 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑘 = 𝑀 → (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁)) = 𝑊) |
10 | 9 | imaeq2d 6080 | . . . . 5 ⊢ (𝑘 = 𝑀 → (𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) = (𝐺 “ 𝑊)) |
11 | 2fveq3 6912 | . . . . 5 ⊢ (𝑘 = 𝑀 → (1st ‘(𝑇‘𝑘)) = (1st ‘(𝑇‘𝑀))) | |
12 | 10, 11 | sseq12d 4029 | . . . 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 24921 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
18 | cvmliftlem.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
19 | 17, 18 | cnf 23270 | . . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
20 | 16, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋) |
21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋) |
22 | 21 | ffund 6741 | . . . 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 35271 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
32 | 21 | fdmd 6747 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → dom 𝐺 = (0[,]1)) |
33 | 31, 32 | sseqtrrd 4037 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ dom 𝐺) |
34 | funfvima2 7251 | . . . 4 ⊢ ((Fun 𝐺 ∧ 𝑊 ⊆ dom 𝐺) → (𝐴 ∈ 𝑊 → (𝐺‘𝐴) ∈ (𝐺 “ 𝑊))) | |
35 | 22, 33, 34 | syl2anc 584 | . . 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 3996 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 ∪ cuni 4912 ∪ ciun 4996 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 0cc0 11153 1c1 11154 − cmin 11490 / cdiv 11918 ℕcn 12264 (,)cioo 13384 [,]cicc 13387 ...cfz 13544 ↾t crest 17467 topGenctg 17484 Cn ccn 23248 Homeochmeo 23777 IIcii 24915 CovMap ccvm 35240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cn 23251 df-ii 24917 |
This theorem is referenced by: cvmliftlem6 35275 cvmliftlem8 35277 cvmliftlem9 35278 |
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