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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 6447 | . . 3 ⊢ ∼ ∈ V |
3 | 2 | ecelqsi 8068 | . 2 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ (𝑊 / ∼ )) |
4 | frgpeccl.w | . . . . . . 7 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | 4 | efgrcl 18479 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
6 | 5 | simpld 490 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V) |
7 | frgp0.m | . . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼) | |
8 | eqid 2825 | . . . . . 6 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
9 | 7, 8, 1 | frgpval 18524 | . . . . 5 ⊢ (𝐼 ∈ V → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
11 | 5 | simprd 491 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
12 | 2on 7835 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7220 | . . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 6, 12, 13 | sylancl 582 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V) |
15 | eqid 2825 | . . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
16 | 8, 15 | frmdbas 17743 | . . . . . 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 2864 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
19 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → ∼ ∈ V) |
20 | fvexd 6448 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
21 | 10, 18, 19, 20 | qusbas 16558 | . . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = (Base‘𝐺)) |
22 | frgpeccl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
23 | 21, 22 | syl6eqr 2879 | . 2 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = 𝐵) |
24 | 3, 23 | eleqtrd 2908 | 1 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3414 I cid 5249 × cxp 5340 Oncon0 5963 ‘cfv 6123 (class class class)co 6905 2oc2o 7820 [cec 8007 / cqs 8008 Word cword 13574 Basecbs 16222 /s cqus 16518 freeMndcfrmd 17738 ~FG cefg 18470 freeGrpcfrgp 18471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-ec 8011 df-qs 8015 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-imas 16521 df-qus 16522 df-frmd 17740 df-frgp 18474 |
This theorem is referenced by: frgpinv 18530 frgpmhm 18531 vrgpf 18534 frgpup3lem 18543 |
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