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Mirrors > Home > MPE Home > Th. List > cayleylem2 | Structured version Visualization version GIF version |
Description: Lemma for cayley 18217. (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 6445 | . . . 4 ⊢ ((𝐹‘𝑥) = (0g‘𝐻) → ((𝐹‘𝑥)‘ 0 ) = ((0g‘𝐻)‘ 0 )) | |
2 | simpr 479 | . . . . . . 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 17837 | . . . . . . . 8 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋) |
6 | 5 | adantr 474 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋) |
7 | cayleylem1.f | . . . . . . . 8 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 7, 3 | grplactval 17904 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = (𝑥 + 0 )) |
9 | 2, 6, 8 | syl2anc 579 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = (𝑥 + 0 )) |
10 | cayleylem1.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
11 | 3, 10, 4 | grprid 17840 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥 + 0 ) = 𝑥) |
12 | 9, 11 | eqtrd 2813 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = 𝑥) |
13 | 3 | fvexi 6460 | . . . . . . . 8 ⊢ 𝑋 ∈ V |
14 | cayleylem1.h | . . . . . . . . 9 ⊢ 𝐻 = (SymGrp‘𝑋) | |
15 | 14 | symgid 18204 | . . . . . . . 8 ⊢ (𝑋 ∈ V → ( I ↾ 𝑋) = (0g‘𝐻)) |
16 | 13, 15 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ 𝑋) = (0g‘𝐻) |
17 | 16 | fveq1i 6447 | . . . . . 6 ⊢ (( I ↾ 𝑋)‘ 0 ) = ((0g‘𝐻)‘ 0 ) |
18 | fvresi 6706 | . . . . . . 7 ⊢ ( 0 ∈ 𝑋 → (( I ↾ 𝑋)‘ 0 ) = 0 ) | |
19 | 6, 18 | syl 17 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (( I ↾ 𝑋)‘ 0 ) = 0 ) |
20 | 17, 19 | syl5eqr 2827 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((0g‘𝐻)‘ 0 ) = 0 ) |
21 | 12, 20 | eqeq12d 2792 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((𝐹‘𝑥)‘ 0 ) = ((0g‘𝐻)‘ 0 ) ↔ 𝑥 = 0 )) |
22 | 1, 21 | syl5ib 236 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (0g‘𝐻) → 𝑥 = 0 )) |
23 | 22 | ralrimiva 3147 | . 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 18215 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
26 | eqid 2777 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | 3, 24, 4, 26 | ghmf1 18073 | . . 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 249 | 1 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 ↦ cmpt 4965 I cid 5260 ↾ cres 5357 –1-1→wf1 6132 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 0gc0g 16486 Grpcgrp 17809 GrpHom cghm 18041 SymGrpcsymg 18180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-tset 16357 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-ghm 18042 df-ga 18106 df-symg 18181 |
This theorem is referenced by: cayley 18217 |
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