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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 6903 | . . . 4 β’ ( βphβπ½) β V | |
4 | ecexg 8709 | . . . 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 8709 | . . . 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 8743 | . . 3 β’ (π = π΄ β [π]( βphβπ½) = [π΄]( βphβπ½)) | |
9 | oveq1 7418 | . . . . 5 β’ (π = π΄ β (π(*πβπ½)πΉ) = (π΄(*πβπ½)πΉ)) | |
10 | 9 | oveq2d 7427 | . . . 4 β’ (π = π΄ β (πΌ(*πβπ½)(π(*πβπ½)πΉ)) = (πΌ(*πβπ½)(π΄(*πβπ½)πΉ))) |
11 | 10 | eceq1d 8744 | . . 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 24800 | . . . 4 β’ (π β πΊ:π΅βΆ(Baseβπ)) |
21 | 20 | ffund 6720 | . . 3 β’ (π β Fun πΊ) |
22 | 2, 5, 7, 8, 11, 21 | fliftval 7315 | . 2 β’ ((π β§ π΄ β βͺ π΅) β (πΊβ[π΄]( βphβπ½)) = [(πΌ(*πβπ½)(π΄(*πβπ½)πΉ))]( βphβπ½)) |
23 | 1, 22 | mpdan 683 | 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 394 = wceq 1539 β wcel 2104 Vcvv 3472 β¨cop 4633 βͺ cuni 4907 β¦ cmpt 5230 ran crn 5676 βcfv 6542 (class class class)co 7411 [cec 8703 0cc0 11112 1c1 11113 Basecbs 17148 TopOnctopon 22632 Cn ccn 22948 IIcii 24615 βphcphtpc 24715 *πcpco 24747 Ο1 cpi1 24750 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-icc 13335 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-qus 17459 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-cn 22951 df-cnp 22952 df-tx 23286 df-hmeo 23479 df-xms 24046 df-ms 24047 df-tms 24048 df-ii 24617 df-htpy 24716 df-phtpy 24717 df-phtpc 24738 df-pco 24752 df-om1 24753 df-pi1 24755 |
This theorem is referenced by: pi1xfr 24802 |
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