![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftpht | Structured version Visualization version GIF version |
Description: If 𝐺 and 𝐻 are path-homotopic, then their lifts 𝑀 and 𝑁 are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
cvmliftpht.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmliftpht.m | ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
cvmliftpht.n | ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) |
cvmliftpht.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmliftpht.p | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
cvmliftpht.e | ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) |
cvmliftpht.g | ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐻) |
Ref | Expression |
---|---|
cvmliftpht | ⊢ (𝜑 → 𝑀( ≃ph‘𝐶)𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftpht.b | . . . 4 ⊢ 𝐵 = ∪ 𝐶 | |
2 | cvmliftpht.m | . . . 4 ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) | |
3 | cvmliftpht.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
4 | cvmliftpht.g | . . . . . 6 ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐻) | |
5 | isphtpc 24373 | . . . . . 6 ⊢ (𝐺( ≃ph‘𝐽)𝐻 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅)) | |
6 | 4, 5 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅)) |
7 | 6 | simp1d 1143 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
8 | cvmliftpht.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
9 | cvmliftpht.e | . . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) | |
10 | 1, 2, 3, 7, 8, 9 | cvmliftiota 33952 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃)) |
11 | 10 | simp1d 1143 | . 2 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐶)) |
12 | cvmliftpht.n | . . . 4 ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃)) | |
13 | 6 | simp2d 1144 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽)) |
14 | phtpc01 24375 | . . . . . . 7 ⊢ (𝐺( ≃ph‘𝐽)𝐻 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1))) | |
15 | 4, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1))) |
16 | 15 | simpld 496 | . . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0)) |
17 | 9, 16 | eqtrd 2773 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0)) |
18 | 1, 12, 3, 13, 8, 17 | cvmliftiota 33952 | . . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃)) |
19 | 18 | simp1d 1143 | . 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶)) |
20 | 6 | simp3d 1145 | . . . 4 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅) |
21 | n0 4307 | . . . 4 ⊢ ((𝐺(PHtpy‘𝐽)𝐻) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) | |
22 | 20, 21 | sylib 217 | . . 3 ⊢ (𝜑 → ∃𝑔 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) |
23 | 3 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
24 | 7, 13 | phtpycn 24362 | . . . . . . 7 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ ((II ×t II) Cn 𝐽)) |
25 | 24 | sselda 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ ((II ×t II) Cn 𝐽)) |
26 | 8 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑃 ∈ 𝐵) |
27 | 9 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (𝐺‘0)) |
28 | 0elunit 13392 | . . . . . . . . 9 ⊢ 0 ∈ (0[,]1) | |
29 | 7 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐺 ∈ (II Cn 𝐽)) |
30 | 13 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐻 ∈ (II Cn 𝐽)) |
31 | simpr 486 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) | |
32 | 29, 30, 31 | phtpyi 24363 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ 0 ∈ (0[,]1)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1))) |
33 | 28, 32 | mpan2 690 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1))) |
34 | 33 | simpld 496 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (0𝑔0) = (𝐺‘0)) |
35 | 27, 34 | eqtr4d 2776 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (0𝑔0)) |
36 | 1, 23, 25, 26, 35 | cvmlift2 33967 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ∃!ℎ ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃)) |
37 | reurex 3356 | . . . . 5 ⊢ (∃!ℎ ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃) → ∃ℎ ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃)) | |
38 | 36, 37 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ∃ℎ ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃)) |
39 | 3 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
40 | 8 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝑃 ∈ 𝐵) |
41 | 9 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (𝐹‘𝑃) = (𝐺‘0)) |
42 | 7 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝐺 ∈ (II Cn 𝐽)) |
43 | 13 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝐻 ∈ (II Cn 𝐽)) |
44 | simplr 768 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) | |
45 | simprl 770 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → ℎ ∈ ((II ×t II) Cn 𝐶)) | |
46 | simprrl 780 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (𝐹 ∘ ℎ) = 𝑔) | |
47 | simprrr 781 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (0ℎ0) = 𝑃) | |
48 | 1, 2, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47 | cvmliftphtlem 33968 | . . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → ℎ ∈ (𝑀(PHtpy‘𝐶)𝑁)) |
49 | 48 | ne0d 4296 | . . . 4 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅) |
50 | 38, 49 | rexlimddv 3155 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅) |
51 | 22, 50 | exlimddv 1939 | . 2 ⊢ (𝜑 → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅) |
52 | isphtpc 24373 | . 2 ⊢ (𝑀( ≃ph‘𝐶)𝑁 ↔ (𝑀 ∈ (II Cn 𝐶) ∧ 𝑁 ∈ (II Cn 𝐶) ∧ (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)) | |
53 | 11, 19, 51, 52 | syl3anbrc 1344 | 1 ⊢ (𝜑 → 𝑀( ≃ph‘𝐶)𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 ∃!wreu 3350 ∅c0 4283 ∪ cuni 4866 class class class wbr 5106 ∘ ccom 5638 ‘cfv 6497 ℩crio 7313 (class class class)co 7358 0cc0 11056 1c1 11057 [,]cicc 13273 Cn ccn 22591 ×t ctx 22927 IIcii 24254 PHtpycphtpy 24347 ≃phcphtpc 24348 CovMap ccvm 33906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-ec 8653 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-mulg 18878 df-cntz 19102 df-cmn 19569 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-cn 22594 df-cnp 22595 df-cmp 22754 df-conn 22779 df-lly 22833 df-nlly 22834 df-tx 22929 df-hmeo 23122 df-xms 23689 df-ms 23690 df-tms 23691 df-ii 24256 df-htpy 24349 df-phtpy 24350 df-phtpc 24371 df-pconn 33872 df-sconn 33873 df-cvm 33907 |
This theorem is referenced by: cvmlift3lem1 33970 |
Copyright terms: Public domain | W3C validator |