cvmliftpht - Mathbox for Mario Carneiro

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Theorem cvmliftpht 34984

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 24936	. . . . . 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 34967	. . 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 24938	. . . . . . 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 2765	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
18	1, 12, 3, 13, 8, 17	cvmliftiota 34967	. . 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 4342	. . . 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 24925	. . . . . . 7 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
25	24	sselda 3972	. . . . . 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 13476	. . . . . . . . 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 24926	. . . . . . . . 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 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (0𝑔0))
36	1, 23, 25, 26, 35	cvmlift2 34982	. . . . 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 34983	. . . . 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 4331	. . . 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 1930	. 2 ⊢ (𝜑 → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
52		isphtpc 24936	. 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 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 ∃!wreu 3362 ∅c0 4318 ∪ cuni 4903 class class class wbr 5143 ∘ ccom 5676 ‘cfv 6542 ℩crio 7370 (class class class)co 7415 0cc0 11136 1c1 11137 [,]cicc 13357 Cn ccn 23144 ×_t ctx 23480 IIcii 24811 PHtpycphtpy 24910 ≃_phcphtpc 24911 CovMap ccvm 34921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-ec 8723 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-cn 23147 df-cnp 23148 df-cmp 23307 df-conn 23332 df-lly 23386 df-nlly 23387 df-tx 23482 df-hmeo 23675 df-xms 24242 df-ms 24243 df-tms 24244 df-ii 24813 df-cncf 24814 df-htpy 24912 df-phtpy 24913 df-phtpc 24934 df-pconn 34887 df-sconn 34888 df-cvm 34922

This theorem is referenced by: cvmlift3lem1 34985

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