cvmlift3lem1 - Mathbox for Mario Carneiro

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Theorem cvmlift3lem1 35270

Description: Lemma for cvmlift3 35279. (Contributed by Mario Carneiro, 6-Jul-2015.)

**Hypotheses**
Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
cvmlift3lem1.1	⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))
cvmlift3lem1.2	⊢ (𝜑 → (𝑀‘0) = 𝑂)
cvmlift3lem1.3	⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))
cvmlift3lem1.4	⊢ (𝜑 → (𝑁‘0) = 𝑂)
cvmlift3lem1.5	⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))

**Assertion**
Ref	Expression
cvmlift3lem1	⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))

Distinct variable groups: 𝑔,𝐽 𝑔,𝐹 𝑔,𝑀 𝑔,𝑁 𝐵,𝑔 𝑔,𝐺 𝐶,𝑔 𝑔,𝐾 𝑃,𝑔 𝑔,𝑂 𝑔,𝑌 Allowed substitution hint: 𝜑(𝑔)

**Proof of Theorem cvmlift3lem1**
Step	Hyp	Ref	Expression
1		cvmlift3.b	. . . 4 ⊢ 𝐵 = ∪ 𝐶
2		eqid 2734	. . . 4 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))
3		eqid 2734	. . . 4 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))
4		cvmlift3.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		cvmlift3.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
6		cvmlift3.e	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
7		cvmlift3lem1.2	. . . . . . 7 ⊢ (𝜑 → (𝑀‘0) = 𝑂)
8	7	fveq2d 6877	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑀‘0)) = (𝐺‘𝑂))
9	6, 8	eqtr4d 2772	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(𝑀‘0)))
10		cvmlift3lem1.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))
11		iiuni 24812	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
12		cvmlift3.y	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝐾
13	11, 12	cnf 23171	. . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐾) → 𝑀:(0[,]1)⟶𝑌)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝑌)
15		0elunit 13476	. . . . . 6 ⊢ 0 ∈ (0[,]1)
16		fvco3 6975	. . . . . 6 ⊢ ((𝑀:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0)))
17	14, 15, 16	sylancl 586	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0)))
18	9, 17	eqtr4d 2772	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ 𝑀)‘0))
19		cvmlift3.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ SConn)
20		cvmlift3lem1.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))
21		cvmlift3lem1.4	. . . . . . 7 ⊢ (𝜑 → (𝑁‘0) = 𝑂)
22	7, 21	eqtr4d 2772	. . . . . 6 ⊢ (𝜑 → (𝑀‘0) = (𝑁‘0))
23		cvmlift3lem1.5	. . . . . 6 ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))
24	19, 10, 20, 22, 23	sconnpht2 35189	. . . . 5 ⊢ (𝜑 → 𝑀( ≃_ph‘𝐾)𝑁)
25		cvmlift3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
26	24, 25	phtpcco2 24937	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝑀)( ≃_ph‘𝐽)(𝐺 ∘ 𝑁))
27	1, 2, 3, 4, 5, 18, 26	cvmliftpht 35269	. . 3 ⊢ (𝜑 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))( ≃_ph‘𝐶)(℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃)))
28		phtpc01 24933	. . 3 ⊢ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))( ≃_ph‘𝐶)(℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃)) → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘0) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘0) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1)))
29	27, 28	syl 17	. 2 ⊢ (𝜑 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘0) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘0) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1)))
30	29	simprd 495	1 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∪ cuni 4881 class class class wbr 5117 ∘ ccom 5656 ⟶wf 6524 ‘cfv 6528 ℩crio 7356 (class class class)co 7400 0cc0 11122 1c1 11123 [,]cicc 13357 Cn ccn 23149 𝑛-Locally cnlly 23390 IIcii 24806 ≃_phcphtpc 24906 PConncpconn 35170 SConncsconn 35171 CovMap ccvm 35206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-inf2 9648 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199 ax-pre-sup 11200 ax-addf 11201

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-tp 4604 df-op 4606 df-uni 4882 df-int 4921 df-iun 4967 df-iin 4968 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7666 df-om 7857 df-1st 7983 df-2nd 7984 df-supp 8155 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-1o 8475 df-2o 8476 df-er 8714 df-ec 8716 df-map 8837 df-ixp 8907 df-en 8955 df-dom 8956 df-sdom 8957 df-fin 8958 df-fsupp 9369 df-fi 9418 df-sup 9449 df-inf 9450 df-oi 9517 df-card 9946 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-div 11888 df-nn 12234 df-2 12296 df-3 12297 df-4 12298 df-5 12299 df-6 12300 df-7 12301 df-8 12302 df-9 12303 df-n0 12495 df-z 12582 df-dec 12702 df-uz 12846 df-q 12958 df-rp 13002 df-xneg 13121 df-xadd 13122 df-xmul 13123 df-ioo 13358 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13662 df-fl 13799 df-seq 14010 df-exp 14070 df-hash 14339 df-cj 15107 df-re 15108 df-im 15109 df-sqrt 15243 df-abs 15244 df-clim 15493 df-sum 15692 df-struct 17153 df-sets 17170 df-slot 17188 df-ndx 17200 df-base 17216 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17503 df-qtop 17508 df-imas 17509 df-xps 17511 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18749 df-mulg 19038 df-cntz 19287 df-cmn 19750 df-psmet 21294 df-xmet 21295 df-met 21296 df-bl 21297 df-mopn 21298 df-cnfld 21303 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-cn 23152 df-cnp 23153 df-cmp 23312 df-conn 23337 df-lly 23391 df-nlly 23392 df-tx 23487 df-hmeo 23680 df-xms 24246 df-ms 24247 df-tms 24248 df-ii 24808 df-cncf 24809 df-htpy 24907 df-phtpy 24908 df-phtpc 24929 df-pco 24943 df-pconn 35172 df-sconn 35173 df-cvm 35207

This theorem is referenced by: cvmlift3lem2 35271

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