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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 6683 | . . . 4 ⊢ ( ≃ph‘𝐽) ∈ V | |
4 | ecexg 8293 | . . . 4 ⊢ (( ≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V) |
6 | ecexg 8293 | . . . 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 8327 | . . 3 ⊢ (𝑔 = 𝐴 → [𝑔]( ≃ph‘𝐽) = [𝐴]( ≃ph‘𝐽)) | |
9 | oveq1 7163 | . . . . 5 ⊢ (𝑔 = 𝐴 → (𝑔(*𝑝‘𝐽)𝐹) = (𝐴(*𝑝‘𝐽)𝐹)) | |
10 | 9 | oveq2d 7172 | . . . 4 ⊢ (𝑔 = 𝐴 → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) = (𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))) |
11 | 10 | eceq1d 8328 | . . 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 23657 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄)) |
21 | 20 | ffund 6518 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
22 | 2, 5, 7, 8, 11, 21 | fliftval 7069 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ∪ 𝐵) → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
23 | 1, 22 | mpdan 685 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ∪ cuni 4838 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 [cec 8287 0cc0 10537 1c1 10538 Basecbs 16483 TopOnctopon 21518 Cn ccn 21832 IIcii 23483 ≃phcphtpc 23573 *𝑝cpco 23604 π1 cpi1 23607 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-qus 16782 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-cn 21835 df-cnp 21836 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-ii 23485 df-htpy 23574 df-phtpy 23575 df-phtpc 23596 df-pco 23609 df-om1 23610 df-pi1 23612 |
This theorem is referenced by: pi1xfr 23659 |
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