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Mirrors > Home > MPE Home > Th. List > cayleylem2 | Structured version Visualization version GIF version |
Description: Lemma for cayley 18778. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
cayleylem1.x | ⊢ 𝑋 = (Base‘𝐺) |
cayleylem1.p | ⊢ + = (+g‘𝐺) |
cayleylem1.u | ⊢ 0 = (0g‘𝐺) |
cayleylem1.h | ⊢ 𝐻 = (SymGrp‘𝑋) |
cayleylem1.s | ⊢ 𝑆 = (Base‘𝐻) |
cayleylem1.f | ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) |
Ref | Expression |
---|---|
cayleylem2 | ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6705 | . . . 4 ⊢ ((𝐹‘𝑥) = (0g‘𝐻) → ((𝐹‘𝑥)‘ 0 ) = ((0g‘𝐻)‘ 0 )) | |
2 | simpr 488 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
3 | cayleylem1.x | . . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺) | |
4 | cayleylem1.u | . . . . . . . . 9 ⊢ 0 = (0g‘𝐺) | |
5 | 3, 4 | grpidcl 18367 | . . . . . . . 8 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋) |
6 | 5 | adantr 484 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋) |
7 | cayleylem1.f | . . . . . . . 8 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 7, 3 | grplactval 18437 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = (𝑥 + 0 )) |
9 | 2, 6, 8 | syl2anc 587 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = (𝑥 + 0 )) |
10 | cayleylem1.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
11 | 3, 10, 4 | grprid 18370 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥 + 0 ) = 𝑥) |
12 | 9, 11 | eqtrd 2774 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = 𝑥) |
13 | 3 | fvexi 6720 | . . . . . . . 8 ⊢ 𝑋 ∈ V |
14 | cayleylem1.h | . . . . . . . . 9 ⊢ 𝐻 = (SymGrp‘𝑋) | |
15 | 14 | symgid 18765 | . . . . . . . 8 ⊢ (𝑋 ∈ V → ( I ↾ 𝑋) = (0g‘𝐻)) |
16 | 13, 15 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ 𝑋) = (0g‘𝐻) |
17 | 16 | fveq1i 6707 | . . . . . 6 ⊢ (( I ↾ 𝑋)‘ 0 ) = ((0g‘𝐻)‘ 0 ) |
18 | fvresi 6977 | . . . . . . 7 ⊢ ( 0 ∈ 𝑋 → (( I ↾ 𝑋)‘ 0 ) = 0 ) | |
19 | 6, 18 | syl 17 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (( I ↾ 𝑋)‘ 0 ) = 0 ) |
20 | 17, 19 | eqtr3id 2788 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((0g‘𝐻)‘ 0 ) = 0 ) |
21 | 12, 20 | eqeq12d 2750 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((𝐹‘𝑥)‘ 0 ) = ((0g‘𝐻)‘ 0 ) ↔ 𝑥 = 0 )) |
22 | 1, 21 | syl5ib 247 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (0g‘𝐻) → 𝑥 = 0 )) |
23 | 22 | ralrimiva 3098 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = (0g‘𝐻) → 𝑥 = 0 )) |
24 | cayleylem1.s | . . . 4 ⊢ 𝑆 = (Base‘𝐻) | |
25 | 3, 10, 4, 14, 24, 7 | cayleylem1 18776 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
26 | eqid 2734 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | 3, 24, 4, 26 | ghmf1 18623 | . . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹:𝑋–1-1→𝑆 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = (0g‘𝐻) → 𝑥 = 0 ))) |
28 | 25, 27 | syl 17 | . 2 ⊢ (𝐺 ∈ Grp → (𝐹:𝑋–1-1→𝑆 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = (0g‘𝐻) → 𝑥 = 0 ))) |
29 | 23, 28 | mpbird 260 | 1 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 Vcvv 3401 ↦ cmpt 5124 I cid 5443 ↾ cres 5542 –1-1→wf1 6366 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 0gc0g 16916 Grpcgrp 18337 GrpHom cghm 18591 SymGrpcsymg 18731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-tset 16786 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-efmnd 18268 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-ghm 18592 df-ga 18656 df-symg 18732 |
This theorem is referenced by: cayley 18778 |
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