cvmlift3lem1 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmlift3lem1		Structured version Visualization version GIF version

Theorem cvmlift3lem1 35324

Description: Lemma for cvmlift3 35333. (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 2737	. . . 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 2737	. . . 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 6910	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑀‘0)) = (𝐺‘𝑂))
9	6, 8	eqtr4d 2780	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(𝑀‘0)))
10		cvmlift3lem1.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))
11		iiuni 24907	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
12		cvmlift3.y	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝐾
13	11, 12	cnf 23254	. . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐾) → 𝑀:(0[,]1)⟶𝑌)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝑌)
15		0elunit 13509	. . . . . 6 ⊢ 0 ∈ (0[,]1)
16		fvco3 7008	. . . . . 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 2780	. . . 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 2780	. . . . . 6 ⊢ (𝜑 → (𝑀‘0) = (𝑁‘0))
23		cvmlift3lem1.5	. . . . . 6 ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))
24	19, 10, 20, 22, 23	sconnpht2 35243	. . . . 5 ⊢ (𝜑 → 𝑀( ≃_ph‘𝐾)𝑁)
25		cvmlift3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
26	24, 25	phtpcco2 25032	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝑀)( ≃_ph‘𝐽)(𝐺 ∘ 𝑁))
27	1, 2, 3, 4, 5, 18, 26	cvmliftpht 35323	. . 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 25028	. . 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 1540 ∈ wcel 2108 ∪ cuni 4907 class class class wbr 5143 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 0cc0 11155 1c1 11156 [,]cicc 13390 Cn ccn 23232 𝑛-Locally cnlly 23473 IIcii 24901 ≃_phcphtpc 25001 PConncpconn 35224 SConncsconn 35225 CovMap ccvm 35260

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-cmp 23395 df-conn 23420 df-lly 23474 df-nlly 23475 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-ii 24903 df-cncf 24904 df-htpy 25002 df-phtpy 25003 df-phtpc 25024 df-pco 25038 df-pconn 35226 df-sconn 35227 df-cvm 35261

This theorem is referenced by: cvmlift3lem2 35325

W3C validator