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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 18886 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
4 | 0ex 5204 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
5 | 4 | prid1 4692 | . . . . . . . 8 ⊢ ∅ ∈ {∅, 1o} |
6 | df2o3 8111 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
7 | 5, 6 | eleqtrri 2912 | . . . . . . 7 ⊢ ∅ ∈ 2o |
8 | opelxpi 5587 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝑗, ∅〉 ∈ (𝐼 × 2o)) | |
9 | 7, 8 | mpan2 689 | . . . . . 6 ⊢ (𝑗 ∈ 𝐼 → 〈𝑗, ∅〉 ∈ (𝐼 × 2o)) |
10 | 9 | adantl 484 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → 〈𝑗, ∅〉 ∈ (𝐼 × 2o)) |
11 | 10 | s1cld 13951 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → 〈“〈𝑗, ∅〉”〉 ∈ Word (𝐼 × 2o)) |
12 | 2on 8105 | . . . . . . 7 ⊢ 2o ∈ On | |
13 | xpexg 7467 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
14 | 12, 13 | mpan2 689 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × 2o) ∈ V) |
15 | 14 | adantr 483 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → (𝐼 × 2o) ∈ V) |
16 | wrdexg 13865 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
17 | fvi 6735 | . . . . 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 2916 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → 〈“〈𝑗, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o))) |
20 | vrgpf.m | . . . 4 ⊢ 𝐺 = (freeGrp‘𝐼) | |
21 | eqid 2821 | . . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o)) | |
22 | vrgpf.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
23 | 20, 1, 21, 22 | frgpeccl 18881 | . . 3 ⊢ (〈“〈𝑗, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o)) → [〈“〈𝑗, ∅〉”〉] ∼ ∈ 𝑋) |
24 | 19, 23 | syl 17 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐼) → [〈“〈𝑗, ∅〉”〉] ∼ ∈ 𝑋) |
25 | 3, 24 | fmpt3d 6875 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 {cpr 4563 〈cop 4567 I cid 5454 × cxp 5548 Oncon0 6186 ⟶wf 6346 ‘cfv 6350 1oc1o 8089 2oc2o 8090 [cec 8281 Word cword 13855 〈“cs1 13943 Basecbs 16477 ~FG cefg 18826 freeGrpcfrgp 18827 varFGrpcvrgp 18828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-s1 13944 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-imas 16775 df-qus 16776 df-frmd 18008 df-frgp 18830 df-vrgp 18831 |
This theorem is referenced by: frgpup3lem 18897 frgpup3 18898 0frgp 18899 frgpnabllem2 18988 frgpnabl 18989 frgpcyg 20714 |
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