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Mirrors > Home > MPE Home > Th. List > frgpeccl | Structured version Visualization version GIF version |
Description: Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
frgp0.m | β’ πΊ = (freeGrpβπΌ) |
frgp0.r | β’ βΌ = ( ~FG βπΌ) |
frgpeccl.w | β’ π = ( I βWord (πΌ Γ 2o)) |
frgpeccl.b | β’ π΅ = (BaseβπΊ) |
Ref | Expression |
---|---|
frgpeccl | β’ (π β π β [π] βΌ β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgp0.r | . . . 4 β’ βΌ = ( ~FG βπΌ) | |
2 | 1 | fvexi 6902 | . . 3 β’ βΌ β V |
3 | 2 | ecelqsi 8763 | . 2 β’ (π β π β [π] βΌ β (π / βΌ )) |
4 | frgpeccl.w | . . . . . . 7 β’ π = ( I βWord (πΌ Γ 2o)) | |
5 | 4 | efgrcl 19577 | . . . . . 6 β’ (π β π β (πΌ β V β§ π = Word (πΌ Γ 2o))) |
6 | 5 | simpld 495 | . . . . 5 β’ (π β π β πΌ β V) |
7 | frgp0.m | . . . . . 6 β’ πΊ = (freeGrpβπΌ) | |
8 | eqid 2732 | . . . . . 6 β’ (freeMndβ(πΌ Γ 2o)) = (freeMndβ(πΌ Γ 2o)) | |
9 | 7, 8, 1 | frgpval 19620 | . . . . 5 β’ (πΌ β V β πΊ = ((freeMndβ(πΌ Γ 2o)) /s βΌ )) |
10 | 6, 9 | syl 17 | . . . 4 β’ (π β π β πΊ = ((freeMndβ(πΌ Γ 2o)) /s βΌ )) |
11 | 5 | simprd 496 | . . . . 5 β’ (π β π β π = Word (πΌ Γ 2o)) |
12 | 2on 8476 | . . . . . . 7 β’ 2o β On | |
13 | xpexg 7733 | . . . . . . 7 β’ ((πΌ β V β§ 2o β On) β (πΌ Γ 2o) β V) | |
14 | 6, 12, 13 | sylancl 586 | . . . . . 6 β’ (π β π β (πΌ Γ 2o) β V) |
15 | eqid 2732 | . . . . . . 7 β’ (Baseβ(freeMndβ(πΌ Γ 2o))) = (Baseβ(freeMndβ(πΌ Γ 2o))) | |
16 | 8, 15 | frmdbas 18729 | . . . . . 6 β’ ((πΌ Γ 2o) β V β (Baseβ(freeMndβ(πΌ Γ 2o))) = Word (πΌ Γ 2o)) |
17 | 14, 16 | syl 17 | . . . . 5 β’ (π β π β (Baseβ(freeMndβ(πΌ Γ 2o))) = Word (πΌ Γ 2o)) |
18 | 11, 17 | eqtr4d 2775 | . . . 4 β’ (π β π β π = (Baseβ(freeMndβ(πΌ Γ 2o)))) |
19 | 2 | a1i 11 | . . . 4 β’ (π β π β βΌ β V) |
20 | fvexd 6903 | . . . 4 β’ (π β π β (freeMndβ(πΌ Γ 2o)) β V) | |
21 | 10, 18, 19, 20 | qusbas 17487 | . . 3 β’ (π β π β (π / βΌ ) = (BaseβπΊ)) |
22 | frgpeccl.b | . . 3 β’ π΅ = (BaseβπΊ) | |
23 | 21, 22 | eqtr4di 2790 | . 2 β’ (π β π β (π / βΌ ) = π΅) |
24 | 3, 23 | eleqtrd 2835 | 1 β’ (π β π β [π] βΌ β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3474 I cid 5572 Γ cxp 5673 Oncon0 6361 βcfv 6540 (class class class)co 7405 2oc2o 8456 [cec 8697 / cqs 8698 Word cword 14460 Basecbs 17140 /s cqus 17447 freeMndcfrmd 18724 ~FG cefg 19568 freeGrpcfrgp 19569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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-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-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 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-imas 17450 df-qus 17451 df-frmd 18726 df-frgp 19572 |
This theorem is referenced by: frgpinv 19626 frgpmhm 19627 vrgpf 19630 frgpup3lem 19639 |
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