cvmliftpht - Mathbox for Mario Carneiro

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Theorem cvmliftpht 35831

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 25190	. . . . . 6 ⊢ (𝐺( ≃_ph‘𝐽)𝐻 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅))
6	4, 5	sylib 221	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅))
7	6	simp1d 1160	. . . 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 35814	. . 3 ⊢ (𝜑 → (𝑀 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑀) = 𝐺 ∧ (𝑀‘0) = 𝑃))
11	10	simp1d 1160	. 2 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐶))
12		cvmliftpht.n	. . . 4 ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))
13	6	simp2d 1161	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
14		phtpc01 25192	. . . . . . 7 ⊢ (𝐺( ≃_ph‘𝐽)𝐻 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
15	4, 14	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝐺‘0) = (𝐻‘0) ∧ (𝐺‘1) = (𝐻‘1)))
16	15	simpld 500	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (𝐻‘0))
17	9, 16	eqtrd 2801	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐻‘0))
18	1, 12, 3, 13, 8, 17	cvmliftiota 35814	. . 3 ⊢ (𝜑 → (𝑁 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑁) = 𝐻 ∧ (𝑁‘0) = 𝑃))
19	18	simp1d 1160	. 2 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐶))
20	6	simp3d 1162	. . . 4 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ≠ ∅)
21		n0 4310	. . . 4 ⊢ ((𝐺(PHtpy‘𝐽)𝐻) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻))
22	20, 21	sylib 221	. . 3 ⊢ (𝜑 → ∃𝑔 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻))
23	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
24	7, 13	phtpycn 25179	. . . . . . 7 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
25	24	sselda 3940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ ((II ×_t II) Cn 𝐽))
26	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑃 ∈ 𝐵)
27	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (𝐺‘0))
28		0elunit 13514	. . . . . . . . 9 ⊢ 0 ∈ (0[,]1)
29	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐺 ∈ (II Cn 𝐽))
30	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝐻 ∈ (II Cn 𝐽))
31		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻))
32	29, 30, 31	phtpyi 25180	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ 0 ∈ (0[,]1)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1)))
33	28, 32	mpan2 704	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ((0𝑔0) = (𝐺‘0) ∧ (1𝑔0) = (𝐺‘1)))
34	33	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (0𝑔0) = (𝐺‘0))
35	27, 34	eqtr4d 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝐹‘𝑃) = (0𝑔0))
36	1, 23, 25, 26, 35	cvmlift2 35829	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ∃!ℎ ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))
37		reurex 3376	. . . . 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 18	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → ∃ℎ ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))
39	3	ad2antrr 739	. . . . . 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 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝑃 ∈ 𝐵)
41	9	ad2antrr 739	. . . . . 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 739	. . . . . 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 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → 𝐻 ∈ (II Cn 𝐽))
44		simplr 781	. . . . . 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 783	. . . . . 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 793	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) ∧ (ℎ ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ ℎ) = 𝑔 ∧ (0ℎ0) = 𝑃))) → (𝐹 ∘ ℎ) = 𝑔)
47		simprrr 794	. . . . . 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 35830	. . . . 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 4298	. . . 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 3175	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐺(PHtpy‘𝐽)𝐻)) → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
51	22, 50	exlimddv 1968	. 2 ⊢ (𝜑 → (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅)
52		isphtpc 25190	. 2 ⊢ (𝑀( ≃_ph‘𝐶)𝑁 ↔ (𝑀 ∈ (II Cn 𝐶) ∧ 𝑁 ∈ (II Cn 𝐶) ∧ (𝑀(PHtpy‘𝐶)𝑁) ≠ ∅))
53	11, 19, 51, 52	syl3anbrc 1362	1 ⊢ (𝜑 → 𝑀( ≃_ph‘𝐶)𝑁)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 ∃!wreu 3370 ∅c0 4289 ∪ cuni 4877 class class class wbr 5114 ∘ ccom 5670 ‘cfv 6543 ℩crio 7379 (class class class)co 7423 0cc0 11118 1c1 11119 [,]cicc 13393 Cn ccn 23418 ×_t ctx 23754 IIcii 25071 PHtpycphtpy 25164 ≃_phcphtpc 25165 CovMap ccvm 35768

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-ec 8705 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-cn 23421 df-cnp 23422 df-cmp 23581 df-conn 23606 df-lly 23660 df-nlly 23661 df-tx 23756 df-hmeo 23949 df-xms 24514 df-ms 24515 df-tms 24516 df-ii 25073 df-cncf 25074 df-htpy 25166 df-phtpy 25167 df-phtpc 25188 df-pconn 35734 df-sconn 35735 df-cvm 35769

This theorem is used by: cvmlift3lem1 35832

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