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Mirrors > Home > MPE Home > Th. List > pi1xfrgim | Structured version Visualization version GIF version |
Description: The mapping 𝐺 between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.) |
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)) |
pi1xfr.i | ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) |
Ref | Expression |
---|---|
pi1xfrgim | ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpIso 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1xfr.p | . . 3 ⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) | |
2 | pi1xfr.q | . . 3 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) | |
3 | pi1xfr.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
4 | pi1xfr.g | . . 3 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)〉) | |
5 | pi1xfr.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
6 | pi1xfr.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
7 | pi1xfr.i | . . 3 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | pi1xfr 23661 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄)) |
9 | eqid 2823 | . . . 4 ⊢ ran (𝑦 ∈ ∪ (Base‘𝑄) ↦ 〈[𝑦]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(𝑦(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) = ran (𝑦 ∈ ∪ (Base‘𝑄) ↦ 〈[𝑦]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(𝑦(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) | |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | pi1xfrcnv 23663 | . . 3 ⊢ (𝜑 → (◡𝐺 = ran (𝑦 ∈ ∪ (Base‘𝑄) ↦ 〈[𝑦]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(𝑦(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∧ ◡𝐺 ∈ (𝑄 GrpHom 𝑃))) |
11 | 10 | simprd 498 | . 2 ⊢ (𝜑 → ◡𝐺 ∈ (𝑄 GrpHom 𝑃)) |
12 | isgim2 18407 | . 2 ⊢ (𝐺 ∈ (𝑃 GrpIso 𝑄) ↔ (𝐺 ∈ (𝑃 GrpHom 𝑄) ∧ ◡𝐺 ∈ (𝑄 GrpHom 𝑃))) | |
13 | 8, 11, 12 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpIso 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 ∪ cuni 4840 ↦ cmpt 5148 ◡ccnv 5556 ran crn 5558 ‘cfv 6357 (class class class)co 7158 [cec 8289 0cc0 10539 1c1 10540 − cmin 10872 [,]cicc 12744 Basecbs 16485 GrpHom cghm 18357 GrpIso cgim 18399 TopOnctopon 21520 Cn ccn 21834 IIcii 23485 ≃phcphtpc 23575 *𝑝cpco 23606 π1 cpi1 23609 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-qus 16784 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-mulg 18227 df-ghm 18358 df-gim 18401 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-cn 21837 df-cnp 21838 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 df-ii 23487 df-htpy 23576 df-phtpy 23577 df-phtpc 23598 df-pco 23611 df-om1 23612 df-pi1 23614 |
This theorem is referenced by: pconnpi1 32486 |
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