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Mirrors > Home > MPE Home > Th. List > pi1xfrval | Structured version Visualization version GIF version |
Description: The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
pi1xfr.p | β’ π = (π½ Ο1 (πΉβ0)) |
pi1xfr.q | β’ π = (π½ Ο1 (πΉβ1)) |
pi1xfr.b | β’ π΅ = (Baseβπ) |
pi1xfr.g | β’ πΊ = ran (π β βͺ π΅ β¦ β¨[π]( βphβπ½), [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½)β©) |
pi1xfr.j | β’ (π β π½ β (TopOnβπ)) |
pi1xfr.f | β’ (π β πΉ β (II Cn π½)) |
pi1xfrval.i | β’ (π β πΌ β (II Cn π½)) |
pi1xfrval.1 | β’ (π β (πΉβ1) = (πΌβ0)) |
pi1xfrval.2 | β’ (π β (πΌβ1) = (πΉβ0)) |
pi1xfrval.a | β’ (π β π΄ β βͺ π΅) |
Ref | Expression |
---|---|
pi1xfrval | β’ (π β (πΊβ[π΄]( βphβπ½)) = [(πΌ(*πβπ½)(π΄(*πβπ½)πΉ))]( βphβπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1xfrval.a | . 2 β’ (π β π΄ β βͺ π΅) | |
2 | pi1xfr.g | . . 3 β’ πΊ = ran (π β βͺ π΅ β¦ β¨[π]( βphβπ½), [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½)β©) | |
3 | fvex 6838 | . . . 4 β’ ( βphβπ½) β V | |
4 | ecexg 8573 | . . . 4 β’ (( βphβπ½) β V β [π]( βphβπ½) β V) | |
5 | 3, 4 | mp1i 13 | . . 3 β’ ((π β§ π β βͺ π΅) β [π]( βphβπ½) β V) |
6 | ecexg 8573 | . . . 4 β’ (( βphβπ½) β V β [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½) β V) | |
7 | 3, 6 | mp1i 13 | . . 3 β’ ((π β§ π β βͺ π΅) β [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½) β V) |
8 | eceq1 8607 | . . 3 β’ (π = π΄ β [π]( βphβπ½) = [π΄]( βphβπ½)) | |
9 | oveq1 7344 | . . . . 5 β’ (π = π΄ β (π(*πβπ½)πΉ) = (π΄(*πβπ½)πΉ)) | |
10 | 9 | oveq2d 7353 | . . . 4 β’ (π = π΄ β (πΌ(*πβπ½)(π(*πβπ½)πΉ)) = (πΌ(*πβπ½)(π΄(*πβπ½)πΉ))) |
11 | 10 | eceq1d 8608 | . . 3 β’ (π = π΄ β [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½) = [(πΌ(*πβπ½)(π΄(*πβπ½)πΉ))]( βphβπ½)) |
12 | pi1xfr.p | . . . . 5 β’ π = (π½ Ο1 (πΉβ0)) | |
13 | pi1xfr.q | . . . . 5 β’ π = (π½ Ο1 (πΉβ1)) | |
14 | pi1xfr.b | . . . . 5 β’ π΅ = (Baseβπ) | |
15 | pi1xfr.j | . . . . 5 β’ (π β π½ β (TopOnβπ)) | |
16 | pi1xfr.f | . . . . 5 β’ (π β πΉ β (II Cn π½)) | |
17 | pi1xfrval.i | . . . . 5 β’ (π β πΌ β (II Cn π½)) | |
18 | pi1xfrval.1 | . . . . 5 β’ (π β (πΉβ1) = (πΌβ0)) | |
19 | pi1xfrval.2 | . . . . 5 β’ (π β (πΌβ1) = (πΉβ0)) | |
20 | 12, 13, 14, 2, 15, 16, 17, 18, 19 | pi1xfrf 24322 | . . . 4 β’ (π β πΊ:π΅βΆ(Baseβπ)) |
21 | 20 | ffund 6655 | . . 3 β’ (π β Fun πΊ) |
22 | 2, 5, 7, 8, 11, 21 | fliftval 7243 | . 2 β’ ((π β§ π΄ β βͺ π΅) β (πΊβ[π΄]( βphβπ½)) = [(πΌ(*πβπ½)(π΄(*πβπ½)πΉ))]( βphβπ½)) |
23 | 1, 22 | mpdan 684 | 1 β’ (π β (πΊβ[π΄]( βphβπ½)) = [(πΌ(*πβπ½)(π΄(*πβπ½)πΉ))]( βphβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 Vcvv 3441 β¨cop 4579 βͺ cuni 4852 β¦ cmpt 5175 ran crn 5621 βcfv 6479 (class class class)co 7337 [cec 8567 0cc0 10972 1c1 10973 Basecbs 17009 TopOnctopon 22165 Cn ccn 22481 IIcii 24144 βphcphtpc 24238 *πcpco 24269 Ο1 cpi1 24272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 ax-mulf 11052 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-2o 8368 df-er 8569 df-ec 8571 df-qs 8575 df-map 8688 df-ixp 8757 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-fi 9268 df-sup 9299 df-inf 9300 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-q 12790 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-ioo 13184 df-icc 13187 df-fz 13341 df-fzo 13484 df-seq 13823 df-exp 13884 df-hash 14146 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-hom 17083 df-cco 17084 df-rest 17230 df-topn 17231 df-0g 17249 df-gsum 17250 df-topgen 17251 df-pt 17252 df-prds 17255 df-xrs 17310 df-qtop 17315 df-imas 17316 df-qus 17317 df-xps 17318 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-mulg 18797 df-cntz 19019 df-cmn 19483 df-psmet 20695 df-xmet 20696 df-met 20697 df-bl 20698 df-mopn 20699 df-cnfld 20704 df-top 22149 df-topon 22166 df-topsp 22188 df-bases 22202 df-cld 22276 df-cn 22484 df-cnp 22485 df-tx 22819 df-hmeo 23012 df-xms 23579 df-ms 23580 df-tms 23581 df-ii 24146 df-htpy 24239 df-phtpy 24240 df-phtpc 24261 df-pco 24274 df-om1 24275 df-pi1 24277 |
This theorem is referenced by: pi1xfr 24324 |
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