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Mirrors > Home > MPE Home > Th. List > vrgpf | Structured version Visualization version GIF version |
Description: The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
vrgpfval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
vrgpfval.u | ⊢ 𝑈 = (varFGrp‘𝐼) |
vrgpf.m | ⊢ 𝐺 = (freeGrp‘𝐼) |
vrgpf.x | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
vrgpf | ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vrgpfval.r | . . 3 ⊢ ∼ = ( ~FG ‘𝐼) | |
2 | vrgpfval.u | . . 3 ⊢ 𝑈 = (varFGrp‘𝐼) | |
3 | 1, 2 | vrgpfval 18884 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
4 | 0ex 5175 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
5 | 4 | prid1 4658 | . . . . . . . 8 ⊢ ∅ ∈ {∅, 1o} |
6 | df2o3 8100 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
7 | 5, 6 | eleqtrri 2889 | . . . . . . 7 ⊢ ∅ ∈ 2o |
8 | opelxpi 5556 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝑗, ∅〉 ∈ (𝐼 × 2o)) | |
9 | 7, 8 | mpan2 690 | . . . . . 6 ⊢ (𝑗 ∈ 𝐼 → 〈𝑗, ∅〉 ∈ (𝐼 × 2o)) |
10 | 9 | adantl 485 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → 〈𝑗, ∅〉 ∈ (𝐼 × 2o)) |
11 | 10 | s1cld 13948 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → 〈“〈𝑗, ∅〉”〉 ∈ Word (𝐼 × 2o)) |
12 | 2on 8094 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7453 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 12, 13 | mpan2 690 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V) |
15 | 14 | adantr 484 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → (𝐼 × 2o) ∈ V) |
16 | wrdexg 13867 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
17 | fvi 6715 | . . . . 5 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) | |
18 | 15, 16, 17 | 3syl 18 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) |
19 | 11, 18 | eleqtrrd 2893 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → 〈“〈𝑗, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o))) |
20 | vrgpf.m | . . . 4 ⊢ 𝐺 = (freeGrp‘𝐼) | |
21 | eqid 2798 | . . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o)) | |
22 | vrgpf.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
23 | 20, 1, 21, 22 | frgpeccl 18879 | . . 3 ⊢ (〈“〈𝑗, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o)) → [〈“〈𝑗, ∅〉”〉] ∼ ∈ 𝑋) |
24 | 19, 23 | syl 17 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → [〈“〈𝑗, ∅〉”〉] ∼ ∈ 𝑋) |
25 | 3, 24 | fmpt3d 6857 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {cpr 4527 〈cop 4531 I cid 5424 × cxp 5517 Oncon0 6159 ⟶wf 6320 ‘cfv 6324 1oc1o 8078 2oc2o 8079 [cec 8270 Word cword 13857 〈“cs1 13940 Basecbs 16475 ~FG cefg 18824 freeGrpcfrgp 18825 varFGrpcvrgp 18826 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-s1 13941 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-imas 16773 df-qus 16774 df-frmd 18006 df-frgp 18828 df-vrgp 18829 |
This theorem is referenced by: frgpup3lem 18895 frgpup3 18896 0frgp 18897 frgpnabllem2 18987 frgpnabl 18988 frgpcyg 20265 |
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