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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 | ⊢ 𝐺 = (freeGrp‘𝐼) |
frgp0.r | ⊢ ∼ = ( ~FG ‘𝐼) |
frgpeccl.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
frgpeccl.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
frgpeccl | ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgp0.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
2 | 1 | fvexi 6677 | . . 3 ⊢ ∼ ∈ V |
3 | 2 | ecelqsi 8342 | . 2 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ (𝑊 / ∼ )) |
4 | frgpeccl.w | . . . . . . 7 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | 4 | efgrcl 18770 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
6 | 5 | simpld 495 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V) |
7 | frgp0.m | . . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼) | |
8 | eqid 2818 | . . . . . 6 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
9 | 7, 8, 1 | frgpval 18813 | . . . . 5 ⊢ (𝐼 ∈ V → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
11 | 5 | simprd 496 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
12 | 2on 8100 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7462 | . . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 6, 12, 13 | sylancl 586 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V) |
15 | eqid 2818 | . . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
16 | 8, 15 | frmdbas 18005 | . . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o)) |
17 | 14, 16 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o)) |
18 | 11, 17 | eqtr4d 2856 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
19 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → ∼ ∈ V) |
20 | fvexd 6678 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
21 | 10, 18, 19, 20 | qusbas 16806 | . . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = (Base‘𝐺)) |
22 | frgpeccl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
23 | 21, 22 | syl6eqr 2871 | . 2 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = 𝐵) |
24 | 3, 23 | eleqtrd 2912 | 1 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 I cid 5452 × cxp 5546 Oncon0 6184 ‘cfv 6348 (class class class)co 7145 2oc2o 8085 [cec 8276 / cqs 8277 Word cword 13849 Basecbs 16471 /s cqus 16766 freeMndcfrmd 18000 ~FG cefg 18761 freeGrpcfrgp 18762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-imas 16769 df-qus 16770 df-frmd 18002 df-frgp 18765 |
This theorem is referenced by: frgpinv 18819 frgpmhm 18820 vrgpf 18823 frgpup3lem 18832 |
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