cvmlift3lem1 - Mathbox for Mario Carneiro

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Theorem cvmlift3lem1 32559

Description: Lemma for cvmlift3 32568. (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 2819	. . . 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 2819	. . . 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 6667	. . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑀‘0)) = (𝐺‘𝑂))
9	6, 8	eqtr4d 2857	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(𝑀‘0)))
10		cvmlift3lem1.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))
11		iiuni 23481	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
12		cvmlift3.y	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝐾
13	11, 12	cnf 21846	. . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐾) → 𝑀:(0[,]1)⟶𝑌)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝑌)
15		0elunit 12847	. . . . . 6 ⊢ 0 ∈ (0[,]1)
16		fvco3 6753	. . . . . 6 ⊢ ((𝑀:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0)))
17	14, 15, 16	sylancl 588	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0)))
18	9, 17	eqtr4d 2857	. . . 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 2857	. . . . . 6 ⊢ (𝜑 → (𝑀‘0) = (𝑁‘0))
23		cvmlift3lem1.5	. . . . . 6 ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))
24	19, 10, 20, 22, 23	sconnpht2 32478	. . . . 5 ⊢ (𝜑 → 𝑀( ≃_ph‘𝐾)𝑁)
25		cvmlift3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
26	24, 25	phtpcco2 23595	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝑀)( ≃_ph‘𝐽)(𝐺 ∘ 𝑁))
27	1, 2, 3, 4, 5, 18, 26	cvmliftpht 32558	. . 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 23592	. . 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 498	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 398 = wceq 1531 ∈ wcel 2108 ∪ cuni 4830 class class class wbr 5057 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 ℩crio 7105 (class class class)co 7148 0cc0 10529 1c1 10530 [,]cicc 12733 Cn ccn 21824 𝑛-Locally cnlly 22065 IIcii 23475 ≃_phcphtpc 23565 PConncpconn 32459 SConncsconn 32460 CovMap ccvm 32495

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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-ec 8283 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-cn 21827 df-cnp 21828 df-cmp 21987 df-conn 22012 df-lly 22066 df-nlly 22067 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-ii 23477 df-htpy 23566 df-phtpy 23567 df-phtpc 23588 df-pco 23601 df-pconn 32461 df-sconn 32462 df-cvm 32496

This theorem is referenced by: cvmlift3lem2 32560

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