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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem14 | Structured version Visualization version GIF version | ||
| Description: Lemma for cvmlift 35304. Putting the results of cvmliftlem11 35300, cvmliftlem13 35301 and cvmliftmo 35289 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 35300 | . . . 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 35301 | . . 3 ⊢ (𝜑 → (𝐾‘0) = 𝑃) |
| 18 | coeq2 5869 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝐾)) | |
| 19 | 18 | eqeq1d 2739 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝐾) = 𝐺)) |
| 20 | fveq1 6905 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (𝑓‘0) = (𝐾‘0)) | |
| 21 | 20 | eqeq1d 2739 | . . . . 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 24907 | . . 3 ⊢ (0[,]1) = ∪ II | |
| 26 | iiconn 24913 | . . . 4 ⊢ II ∈ Conn | |
| 27 | 26 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ Conn) |
| 28 | iinllyconn 35259 | . . . 4 ⊢ II ∈ 𝑛-Locally Conn | |
| 29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ 𝑛-Locally Conn) |
| 30 | 0elunit 13509 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
| 31 | 30 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0[,]1)) |
| 32 | 2, 25, 4, 27, 29, 31, 5, 6, 7 | cvmliftmo 35289 | . 2 ⊢ (𝜑 → ∃*𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
| 33 | reu5 3382 | . 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 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 ∃*wrmo 3379 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 〈cop 4632 ∪ cuni 4907 ∪ ciun 4991 ↦ cmpt 5225 I cid 5577 × cxp 5683 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 0cc0 11155 1c1 11156 − cmin 11492 / cdiv 11920 ℕcn 12266 (,)cioo 13387 [,]cicc 13390 ...cfz 13547 seqcseq 14042 ↾t crest 17465 topGenctg 17482 Cn ccn 23232 Conncconn 23419 𝑛-Locally cnlly 23473 Homeochmeo 23761 IIcii 24901 CovMap ccvm 35260 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-nei 23106 df-cn 23235 df-cnp 23236 df-conn 23420 df-lly 23474 df-nlly 23475 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-ii 24903 df-cncf 24904 df-htpy 25002 df-phtpy 25003 df-phtpc 25024 df-pconn 35226 df-sconn 35227 df-cvm 35261 |
| This theorem is referenced by: cvmliftlem15 35303 |
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