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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 | ⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) |
pi1xfr.q | ⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) |
pi1xfr.b | ⊢ 𝐵 = (Base‘𝑃) |
pi1xfr.g | ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)〉) |
pi1xfr.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
pi1xfr.f | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
pi1xfrval.i | ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
pi1xfrval.1 | ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0)) |
pi1xfrval.2 | ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
pi1xfrval.a | ⊢ (𝜑 → 𝐴 ∈ ∪ 𝐵) |
Ref | Expression |
---|---|
pi1xfrval | ⊢ (𝜑 → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1xfrval.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝐵) | |
2 | pi1xfr.g | . . 3 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)〉) | |
3 | fvex 6739 | . . . 4 ⊢ ( ≃ph‘𝐽) ∈ V | |
4 | ecexg 8404 | . . . 4 ⊢ (( ≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V) |
6 | ecexg 8404 | . . . 4 ⊢ (( ≃ph‘𝐽) ∈ V → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V) | |
7 | 3, 6 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V) |
8 | eceq1 8438 | . . 3 ⊢ (𝑔 = 𝐴 → [𝑔]( ≃ph‘𝐽) = [𝐴]( ≃ph‘𝐽)) | |
9 | oveq1 7229 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔(*𝑝‘𝐽)𝐹) = (𝐴(*𝑝‘𝐽)𝐹)) | |
10 | 9 | oveq2d 7238 | . . . 4 ⊢ (𝑔 = 𝐴 → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) = (𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))) |
11 | 10 | eceq1d 8439 | . . 3 ⊢ (𝑔 = 𝐴 → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
12 | pi1xfr.p | . . . . 5 ⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) | |
13 | pi1xfr.q | . . . . 5 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) | |
14 | pi1xfr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
15 | pi1xfr.j | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
16 | pi1xfr.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
17 | pi1xfrval.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) | |
18 | pi1xfrval.1 | . . . . 5 ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0)) | |
19 | pi1xfrval.2 | . . . . 5 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) | |
20 | 12, 13, 14, 2, 15, 16, 17, 18, 19 | pi1xfrf 23963 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄)) |
21 | 20 | ffund 6558 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
22 | 2, 5, 7, 8, 11, 21 | fliftval 7134 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ∪ 𝐵) → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
23 | 1, 22 | mpdan 687 | 1 ⊢ (𝜑 → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3415 〈cop 4556 ∪ cuni 4828 ↦ cmpt 5144 ran crn 5561 ‘cfv 6389 (class class class)co 7222 [cec 8398 0cc0 10742 1c1 10743 Basecbs 16773 TopOnctopon 21820 Cn ccn 22134 IIcii 23785 ≃phcphtpc 23879 *𝑝cpco 23910 π1 cpi1 23913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 ax-pre-sup 10820 ax-mulf 10822 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-iin 4916 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-se 5519 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-isom 6398 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-of 7478 df-om 7654 df-1st 7770 df-2nd 7771 df-supp 7913 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-2o 8212 df-er 8400 df-ec 8402 df-qs 8406 df-map 8519 df-ixp 8588 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-fsupp 8999 df-fi 9040 df-sup 9071 df-inf 9072 df-oi 9139 df-card 9568 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-div 11503 df-nn 11844 df-2 11906 df-3 11907 df-4 11908 df-5 11909 df-6 11910 df-7 11911 df-8 11912 df-9 11913 df-n0 12104 df-z 12190 df-dec 12307 df-uz 12452 df-q 12558 df-rp 12600 df-xneg 12717 df-xadd 12718 df-xmul 12719 df-ioo 12952 df-icc 12955 df-fz 13109 df-fzo 13252 df-seq 13588 df-exp 13649 df-hash 13910 df-cj 14675 df-re 14676 df-im 14677 df-sqrt 14811 df-abs 14812 df-struct 16713 df-sets 16730 df-slot 16748 df-ndx 16758 df-base 16774 df-ress 16798 df-plusg 16828 df-mulr 16829 df-starv 16830 df-sca 16831 df-vsca 16832 df-ip 16833 df-tset 16834 df-ple 16835 df-ds 16837 df-unif 16838 df-hom 16839 df-cco 16840 df-rest 16940 df-topn 16941 df-0g 16959 df-gsum 16960 df-topgen 16961 df-pt 16962 df-prds 16965 df-xrs 17020 df-qtop 17025 df-imas 17026 df-qus 17027 df-xps 17028 df-mre 17102 df-mrc 17103 df-acs 17105 df-mgm 18127 df-sgrp 18176 df-mnd 18187 df-submnd 18232 df-mulg 18502 df-cntz 18724 df-cmn 19185 df-psmet 20368 df-xmet 20369 df-met 20370 df-bl 20371 df-mopn 20372 df-cnfld 20377 df-top 21804 df-topon 21821 df-topsp 21843 df-bases 21856 df-cld 21929 df-cn 22137 df-cnp 22138 df-tx 22472 df-hmeo 22665 df-xms 23231 df-ms 23232 df-tms 23233 df-ii 23787 df-htpy 23880 df-phtpy 23881 df-phtpc 23902 df-pco 23915 df-om1 23916 df-pi1 23918 |
This theorem is referenced by: pi1xfr 23965 |
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