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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 6934 | . . 3 ⊢ ∼ ∈ V |
3 | 2 | ecelqsi 8831 | . 2 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ (𝑊 / ∼ )) |
4 | frgpeccl.w | . . . . . . 7 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | 4 | efgrcl 19757 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
6 | 5 | simpld 494 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V) |
7 | frgp0.m | . . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼) | |
8 | eqid 2740 | . . . . . 6 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
9 | 7, 8, 1 | frgpval 19800 | . . . . 5 ⊢ (𝐼 ∈ V → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
11 | 5 | simprd 495 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
12 | 2on 8536 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7785 | . . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 6, 12, 13 | sylancl 585 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V) |
15 | eqid 2740 | . . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
16 | 8, 15 | frmdbas 18887 | . . . . . 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 2783 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
19 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → ∼ ∈ V) |
20 | fvexd 6935 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
21 | 10, 18, 19, 20 | qusbas 17605 | . . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = (Base‘𝐺)) |
22 | frgpeccl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
23 | 21, 22 | eqtr4di 2798 | . 2 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = 𝐵) |
24 | 3, 23 | eleqtrd 2846 | 1 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 I cid 5592 × cxp 5698 Oncon0 6395 ‘cfv 6573 (class class class)co 7448 2oc2o 8516 [cec 8761 / cqs 8762 Word cword 14562 Basecbs 17258 /s cqus 17565 freeMndcfrmd 18882 ~FG cefg 19748 freeGrpcfrgp 19749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-imas 17568 df-qus 17569 df-frmd 18884 df-frgp 19752 |
This theorem is referenced by: frgpinv 19806 frgpmhm 19807 vrgpf 19810 frgpup3lem 19819 |
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