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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem14 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 35284. Putting the results of cvmliftlem11 35280, cvmliftlem13 35281 and cvmliftmo 35269 together, we have that 𝐾 is a continuous function, satisfies 𝐹 ∘ 𝐾 = 𝐺 and 𝐾(0) = 𝑃, and is equal to any other function which also has these properties, so it follows that 𝐾 is the unique lift of 𝐺. (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 (,)) |
cvmliftlem.q | ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})) |
cvmliftlem.k | ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) |
Ref | Expression |
---|---|
cvmliftlem14 | ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftlem.1 | . . . . 5 ⊢ 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))))}) | |
2 | cvmliftlem.b | . . . . 5 ⊢ 𝐵 = ∪ 𝐶 | |
3 | cvmliftlem.x | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | cvmliftlem.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
5 | cvmliftlem.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
6 | cvmliftlem.p | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
7 | cvmliftlem.e | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) | |
8 | cvmliftlem.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | cvmliftlem.t | . . . . 5 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) | |
10 | cvmliftlem.a | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) | |
11 | cvmliftlem.l | . . . . 5 ⊢ 𝐿 = (topGen‘ran (,)) | |
12 | cvmliftlem.q | . . . . 5 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})) | |
13 | cvmliftlem.k | . . . . 5 ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | cvmliftlem11 35280 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺)) |
15 | 14 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐶)) |
16 | 14 | simprd 495 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺) |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | cvmliftlem13 35281 | . . 3 ⊢ (𝜑 → (𝐾‘0) = 𝑃) |
18 | coeq2 5872 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝐾)) | |
19 | 18 | eqeq1d 2737 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝐾) = 𝐺)) |
20 | fveq1 6906 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (𝑓‘0) = (𝐾‘0)) | |
21 | 20 | eqeq1d 2737 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((𝑓‘0) = 𝑃 ↔ (𝐾‘0) = 𝑃)) |
22 | 19, 21 | anbi12d 632 | . . . 4 ⊢ (𝑓 = 𝐾 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ ((𝐹 ∘ 𝐾) = 𝐺 ∧ (𝐾‘0) = 𝑃))) |
23 | 22 | rspcev 3622 | . . 3 ⊢ ((𝐾 ∈ (II Cn 𝐶) ∧ ((𝐹 ∘ 𝐾) = 𝐺 ∧ (𝐾‘0) = 𝑃)) → ∃𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
24 | 15, 16, 17, 23 | syl12anc 837 | . 2 ⊢ (𝜑 → ∃𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
25 | iiuni 24921 | . . 3 ⊢ (0[,]1) = ∪ II | |
26 | iiconn 24927 | . . . 4 ⊢ II ∈ Conn | |
27 | 26 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ Conn) |
28 | iinllyconn 35239 | . . . 4 ⊢ II ∈ 𝑛-Locally Conn | |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ 𝑛-Locally Conn) |
30 | 0elunit 13506 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
31 | 30 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0[,]1)) |
32 | 2, 25, 4, 27, 29, 31, 5, 6, 7 | cvmliftmo 35269 | . 2 ⊢ (𝜑 → ∃*𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
33 | reu5 3380 | . 2 ⊢ (∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ↔ (∃𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃) ∧ ∃*𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))) | |
34 | 24, 32, 33 | sylanbrc 583 | 1 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 ∃*wrmo 3377 {crab 3433 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 〈cop 4637 ∪ cuni 4912 ∪ ciun 4996 ↦ cmpt 5231 I cid 5582 × cxp 5687 ◡ccnv 5688 ran crn 5690 ↾ cres 5691 “ cima 5692 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 0cc0 11153 1c1 11154 − cmin 11490 / cdiv 11918 ℕcn 12264 (,)cioo 13384 [,]cicc 13387 ...cfz 13544 seqcseq 14039 ↾t crest 17467 topGenctg 17484 Cn ccn 23248 Conncconn 23435 𝑛-Locally cnlly 23489 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-rep 5285 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 ax-addf 11232 |
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-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 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-se 5642 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 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-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-nei 23122 df-cn 23251 df-cnp 23252 df-conn 23436 df-lly 23490 df-nlly 23491 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-ii 24917 df-cncf 24918 df-htpy 25016 df-phtpy 25017 df-phtpc 25038 df-pconn 35206 df-sconn 35207 df-cvm 35241 |
This theorem is referenced by: cvmliftlem15 35283 |
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