cvmliftpht - Mathbox for Mario Carneiro

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Theorem cvmliftpht 33280

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
cvmliftpht	⊢ (𝜑 → 𝑀( ≃_ph‘𝐶)𝑁)

Distinct variable groups: 𝐵,𝑓 𝑓,𝐹 𝑓,𝐽 𝐶,𝑓 𝑓,𝐺 𝑓,𝐻 𝑃,𝑓 Allowed substitution hints: 𝜑(𝑓) 𝑀(𝑓) 𝑁(𝑓)

Proof of Theorem cvmliftpht Dummy variables ℎ 𝑔 are mutually distinct and distinct from all other variables.

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 24157	. . . . . 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 1141	. . . 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 33263	. . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
11	10	simp1d 1141	. 2 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐶))
12		cvmliftpht.n	. . . 4 ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
13	6	simp2d 1142	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
14		phtpc01 24159	. . . . . . 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 495	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0))
17	9, 16	eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
18	1, 12, 3, 13, 8, 17	cvmliftiota 33263	. . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
19	18	simp1d 1141	. 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶))
20	6	simp3d 1143	. . . 4 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅)
21		n0 4280	. . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
24	7, 13	phtpycn 24146	. . . . . . 7 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
25	24	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ ((II ×_t II) Cn 𝐽))
26	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑃 ∈ 𝐵)
27	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (𝐺‘0))
28		0elunit 13201	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
29	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐺 ∈ (II Cn 𝐽))
30	13	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐻 ∈ (II Cn 𝐽))
31		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻))
32	29, 30, 31	phtpyi 24147	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ 0 ∈ (0[,]1)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1)))
33	28, 32	mpan2 688	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1)))
34	33	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (0𝑔0) = (𝐺‘0))
35	27, 34	eqtr4d 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (0𝑔0))
36	1, 23, 25, 26, 35	cvmlift2 33278	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ∃!ℎ ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))
37		reurex 3362	. . . . 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 723	. . . . . 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 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝑃 ∈ 𝐵)
41	9	ad2antrr 723	. . . . . 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 723	. . . . . 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 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝐻 ∈ (II Cn 𝐽))
44		simplr 766	. . . . . 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 768	. . . . . 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 778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (𝐹 ∘ ℎ) = 𝑔)
47		simprrr 779	. . . . . 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 33279	. . . . 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 4269	. . . 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 3220	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
51	22, 50	exlimddv 1938	. 2 ⊢ (𝜑 → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
52		isphtpc 24157	. 2 ⊢ (𝑀( ≃_ph‘𝐶)𝑁 ↔ (𝑀 ∈ (II Cn 𝐶) ∧ 𝑁 ∈ (II Cn 𝐶) ∧ (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅))
53	11, 19, 51, 52	syl3anbrc 1342	1 ⊢ (𝜑 → 𝑀( ≃_ph‘𝐶)𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∃!wreu 3066 ∅c0 4256 ∪ cuni 4839 class class class wbr 5074 ∘ ccom 5593 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 0cc0 10871 1c1 10872 [,]cicc 13082 Cn ccn 22375 ×_t ctx 22711 IIcii 24038 PHtpycphtpy 24131 ≃_phcphtpc 24132 CovMap ccvm 33217

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-ec 8500 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-cmp 22538 df-conn 22563 df-lly 22617 df-nlly 22618 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-ii 24040 df-htpy 24133 df-phtpy 24134 df-phtpc 24155 df-pconn 33183 df-sconn 33184 df-cvm 33218

This theorem is referenced by: cvmlift3lem1 33281

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