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Mirrors > Home > MPE Home > Th. List > pi1bas | Structured version Visualization version GIF version |
Description: The base set of the fundamental group of a topological space at a given base point. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
pi1val.g | ⊢ 𝐺 = (𝐽 π1 𝑌) |
pi1val.1 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
pi1val.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
pi1val.o | ⊢ 𝑂 = (𝐽 Ω1 𝑌) |
pi1bas.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
pi1bas.k | ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) |
Ref | Expression |
---|---|
pi1bas | ⊢ (𝜑 → 𝐵 = (𝐾 / ( ≃ph‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1val.g | . . . 4 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
2 | pi1val.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | pi1val.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
4 | pi1val.o | . . . 4 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
5 | 1, 2, 3, 4 | pi1val 22837 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽))) |
6 | eqidd 2623 | . . 3 ⊢ (𝜑 → (Base‘𝑂) = (Base‘𝑂)) | |
7 | fvexd 6203 | . . 3 ⊢ (𝜑 → ( ≃ph‘𝐽) ∈ V) | |
8 | ovex 6678 | . . . . 5 ⊢ (𝐽 Ω1 𝑌) ∈ V | |
9 | 4, 8 | eqeltri 2697 | . . . 4 ⊢ 𝑂 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑂 ∈ V) |
11 | 5, 6, 7, 10 | qusbas 16205 | . 2 ⊢ (𝜑 → ((Base‘𝑂) / ( ≃ph‘𝐽)) = (Base‘𝐺)) |
12 | pi1bas.k | . . 3 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) | |
13 | qseq1 7796 | . . 3 ⊢ (𝐾 = (Base‘𝑂) → (𝐾 / ( ≃ph‘𝐽)) = ((Base‘𝑂) / ( ≃ph‘𝐽))) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝜑 → (𝐾 / ( ≃ph‘𝐽)) = ((Base‘𝑂) / ( ≃ph‘𝐽))) |
15 | pi1bas.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
16 | 11, 14, 15 | 3eqtr4rd 2667 | 1 ⊢ (𝜑 → 𝐵 = (𝐾 / ( ≃ph‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ‘cfv 5888 (class class class)co 6650 / cqs 7741 Basecbs 15857 TopOnctopon 20715 ≃phcphtpc 22768 Ω1 comi 22801 π1 cpi1 22803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-imas 16168 df-qus 16169 df-topon 20716 df-pi1 22808 |
This theorem is referenced by: pi1buni 22840 pi1bas2 22841 |
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