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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 19800 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
4 | 3 | fveq2d 6911 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ )) |
5 | simpr 484 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼) | |
6 | 0ex 5313 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4767 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8513 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2838 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | opelxpi 5726 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) | |
11 | 5, 9, 10 | sylancl 586 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) |
12 | 11 | s1cld 14638 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2o)) |
13 | simpl 482 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉) | |
14 | 2on 8519 | . . . . . 6 ⊢ 2o ∈ On | |
15 | xpexg 7769 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
16 | 13, 14, 15 | sylancl 586 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2o) ∈ V) |
17 | wrdexg 14559 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
18 | fvi 6985 | . . . . 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 2842 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o))) |
21 | eqid 2735 | . . . 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 2735 | . . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) = (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) | |
25 | 21, 22, 1, 23, 24 | frgpinv 19797 | . . 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 14800 | . . . . . 6 ⊢ (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉 | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉) |
29 | 28 | coeq2d 5876 | . . . 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 19746 | . . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉):(𝐼 × 2o)⟶(𝐼 × 2o) |
31 | s1co 14869 | . . . . 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 586 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) |
33 | 24 | efgmval 19745 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) |
34 | 5, 9, 33 | sylancl 586 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) |
35 | df-ov 7434 | . . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) | |
36 | dif0 4384 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
37 | 36 | opeq2i 4882 | . . . . . 6 ⊢ 〈𝐴, (1o ∖ ∅)〉 = 〈𝐴, 1o〉 |
38 | 34, 35, 37 | 3eqtr3g 2798 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) = 〈𝐴, 1o〉) |
39 | 38 | s1eqd 14636 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉 = 〈“〈𝐴, 1o〉”〉) |
40 | 29, 32, 39 | 3eqtrd 2779 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = 〈“〈𝐴, 1o〉”〉) |
41 | 40 | eceq1d 8784 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ = [〈“〈𝐴, 1o〉”〉] ∼ ) |
42 | 4, 26, 41 | 3eqtrd 2779 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 {cpr 4633 〈cop 4637 I cid 5582 × cxp 5687 ∘ ccom 5693 Oncon0 6386 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1oc1o 8498 2oc2o 8499 [cec 8742 Word cword 14549 〈“cs1 14630 reversecreverse 14793 invgcminusg 18965 ~FG cefg 19739 freeGrpcfrgp 19740 varFGrpcvrgp 19741 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-frmd 18875 df-grp 18967 df-minusg 18968 df-efg 19742 df-frgp 19743 df-vrgp 19744 |
This theorem is referenced by: frgpup3lem 19810 |
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