cvmliftpht - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftpht		Structured version Visualization version GIF version

Theorem cvmliftpht 35146

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 25011	. . . . . 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 1139	. . . 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 35129	. . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
11	10	simp1d 1139	. 2 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐶))
12		cvmliftpht.n	. . . 4 ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
13	6	simp2d 1140	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
14		phtpc01 25013	. . . . . . 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 493	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0))
17	9, 16	eqtrd 2766	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
18	1, 12, 3, 13, 8, 17	cvmliftiota 35129	. . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
19	18	simp1d 1139	. 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶))
20	6	simp3d 1141	. . . 4 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅)
21		n0 4349	. . . 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 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
24	7, 13	phtpycn 25000	. . . . . . 7 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
25	24	sselda 3979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ ((II ×_t II) Cn 𝐽))
26	8	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑃 ∈ 𝐵)
27	9	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (𝐺‘0))
28		0elunit 13500	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
29	7	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐺 ∈ (II Cn 𝐽))
30	13	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐻 ∈ (II Cn 𝐽))
31		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻))
32	29, 30, 31	phtpyi 25001	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ 0 ∈ (0[,]1)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1)))
33	28, 32	mpan2 689	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1)))
34	33	simpld 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (0𝑔0) = (𝐺‘0))
35	27, 34	eqtr4d 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (0𝑔0))
36	1, 23, 25, 26, 35	cvmlift2 35144	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ∃!ℎ ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))
37		reurex 3368	. . . . 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 724	. . . . . 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 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝑃 ∈ 𝐵)
41	9	ad2antrr 724	. . . . . 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 724	. . . . . 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 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝐻 ∈ (II Cn 𝐽))
44		simplr 767	. . . . . 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 769	. . . . . 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 779	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (𝐹 ∘ ℎ) = 𝑔)
47		simprrr 780	. . . . . 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 35145	. . . . 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 4338	. . . 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 3151	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
51	22, 50	exlimddv 1931	. 2 ⊢ (𝜑 → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
52		isphtpc 25011	. 2 ⊢ (𝑀( ≃_ph‘𝐶)𝑁 ↔ (𝑀 ∈ (II Cn 𝐶) ∧ 𝑁 ∈ (II Cn 𝐶) ∧ (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅))
53	11, 19, 51, 52	syl3anbrc 1340	1 ⊢ (𝜑 → 𝑀( ≃_ph‘𝐶)𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ∃!wreu 3362 ∅c0 4325 ∪ cuni 4913 class class class wbr 5153 ∘ ccom 5686 ‘cfv 6554 ℩crio 7379 (class class class)co 7424 0cc0 11158 1c1 11159 [,]cicc 13381 Cn ccn 23219 ×_t ctx 23555 IIcii 24886 PHtpycphtpy 24985 ≃_phcphtpc 24986 CovMap ccvm 35083

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-ec 8736 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 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 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-cn 23222 df-cnp 23223 df-cmp 23382 df-conn 23407 df-lly 23461 df-nlly 23462 df-tx 23557 df-hmeo 23750 df-xms 24317 df-ms 24318 df-tms 24319 df-ii 24888 df-cncf 24889 df-htpy 24987 df-phtpy 24988 df-phtpc 25009 df-pconn 35049 df-sconn 35050 df-cvm 35084

This theorem is referenced by: cvmlift3lem1 35147

W3C validator