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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 6690 | . . 3 ⊢ ∼ ∈ V |
3 | 2 | ecelqsi 8386 | . 2 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ (𝑊 / ∼ )) |
4 | frgpeccl.w | . . . . . . 7 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | 4 | efgrcl 18961 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
6 | 5 | simpld 498 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V) |
7 | frgp0.m | . . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼) | |
8 | eqid 2738 | . . . . . 6 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
9 | 7, 8, 1 | frgpval 19004 | . . . . 5 ⊢ (𝐼 ∈ V → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
10 | 6, 9 | syl 17 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
11 | 5 | simprd 499 | . . . . 5 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
12 | 2on 8141 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7493 | . . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 6, 12, 13 | sylancl 589 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V) |
15 | eqid 2738 | . . . . . . 7 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
16 | 8, 15 | frmdbas 18135 | . . . . . 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 2776 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
19 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → ∼ ∈ V) |
20 | fvexd 6691 | . . . 4 ⊢ (𝑋 ∈ 𝑊 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
21 | 10, 18, 19, 20 | qusbas 16923 | . . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = (Base‘𝐺)) |
22 | frgpeccl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
23 | 21, 22 | eqtr4di 2791 | . 2 ⊢ (𝑋 ∈ 𝑊 → (𝑊 / ∼ ) = 𝐵) |
24 | 3, 23 | eleqtrd 2835 | 1 ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3398 I cid 5428 × cxp 5523 Oncon0 6172 ‘cfv 6339 (class class class)co 7172 2oc2o 8127 [cec 8320 / cqs 8321 Word cword 13957 Basecbs 16588 /s cqus 16883 freeMndcfrmd 18130 ~FG cefg 18952 freeGrpcfrgp 18953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-2o 8134 df-er 8322 df-ec 8324 df-qs 8328 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-sup 8981 df-inf 8982 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-fz 12984 df-fzo 13127 df-hash 13785 df-word 13958 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-plusg 16683 df-mulr 16684 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-imas 16886 df-qus 16887 df-frmd 18132 df-frgp 18956 |
This theorem is referenced by: frgpinv 19010 frgpmhm 19011 vrgpf 19014 frgpup3lem 19023 |
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