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Mirrors > Home > MPE Home > Th. List > vrgpinv | Structured version Visualization version GIF version |
Description: The inverse of a generating element is represented by 〈𝐴, 1〉 instead of 〈𝐴, 0〉. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
vrgpfval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
vrgpfval.u | ⊢ 𝑈 = (varFGrp‘𝐼) |
vrgpf.m | ⊢ 𝐺 = (freeGrp‘𝐼) |
vrgpinv.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
vrgpinv | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vrgpfval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
2 | vrgpfval.u | . . . 4 ⊢ 𝑈 = (varFGrp‘𝐼) | |
3 | 1, 2 | vrgpval 18896 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
4 | 3 | fveq2d 6677 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ )) |
5 | simpr 487 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼) | |
6 | 0ex 5214 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4701 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8120 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2915 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | opelxpi 5595 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) | |
11 | 5, 9, 10 | sylancl 588 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) |
12 | 11 | s1cld 13960 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2o)) |
13 | simpl 485 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉) | |
14 | 2on 8114 | . . . . . 6 ⊢ 2o ∈ On | |
15 | xpexg 7476 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
16 | 13, 14, 15 | sylancl 588 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2o) ∈ V) |
17 | wrdexg 13874 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
18 | fvi 6743 | . . . . 5 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) | |
19 | 16, 17, 18 | 3syl 18 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) |
20 | 12, 19 | eleqtrrd 2919 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o))) |
21 | eqid 2824 | . . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o)) | |
22 | vrgpf.m | . . . 4 ⊢ 𝐺 = (freeGrp‘𝐼) | |
23 | vrgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
24 | eqid 2824 | . . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) = (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) | |
25 | 21, 22, 1, 23, 24 | frgpinv 18893 | . . 3 ⊢ (〈“〈𝐴, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o)) → (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ ) = [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ ) |
26 | 20, 25 | syl 17 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ ) = [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ ) |
27 | revs1 14130 | . . . . . 6 ⊢ (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉 | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉) |
29 | 28 | coeq2d 5736 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉)) |
30 | 24 | efgmf 18842 | . . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉):(𝐼 × 2o)⟶(𝐼 × 2o) |
31 | s1co 14198 | . . . . 5 ⊢ ((〈𝐴, ∅〉 ∈ (𝐼 × 2o) ∧ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉):(𝐼 × 2o)⟶(𝐼 × 2o)) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) | |
32 | 11, 30, 31 | sylancl 588 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) |
33 | 24 | efgmval 18841 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) |
34 | 5, 9, 33 | sylancl 588 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) |
35 | df-ov 7162 | . . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) | |
36 | dif0 4335 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
37 | 36 | opeq2i 4810 | . . . . . 6 ⊢ 〈𝐴, (1o ∖ ∅)〉 = 〈𝐴, 1o〉 |
38 | 34, 35, 37 | 3eqtr3g 2882 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) = 〈𝐴, 1o〉) |
39 | 38 | s1eqd 13958 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉 = 〈“〈𝐴, 1o〉”〉) |
40 | 29, 32, 39 | 3eqtrd 2863 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = 〈“〈𝐴, 1o〉”〉) |
41 | 40 | eceq1d 8331 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ = [〈“〈𝐴, 1o〉”〉] ∼ ) |
42 | 4, 26, 41 | 3eqtrd 2863 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∖ cdif 3936 ∅c0 4294 {cpr 4572 〈cop 4576 I cid 5462 × cxp 5556 ∘ ccom 5562 Oncon0 6194 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1oc1o 8098 2oc2o 8099 [cec 8290 Word cword 13864 〈“cs1 13952 reversecreverse 14123 invgcminusg 18107 ~FG cefg 18835 freeGrpcfrgp 18836 varFGrpcvrgp 18837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-ec 8294 df-qs 8298 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-lsw 13918 df-concat 13926 df-s1 13953 df-substr 14006 df-pfx 14036 df-splice 14115 df-reverse 14124 df-s2 14213 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-0g 16718 df-imas 16784 df-qus 16785 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-frmd 18017 df-grp 18109 df-minusg 18110 df-efg 18838 df-frgp 18839 df-vrgp 18840 |
This theorem is referenced by: frgpup3lem 18906 |
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