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Mirrors > Home > MPE Home > Th. List > cayleylem2 | Structured version Visualization version GIF version |
Description: Lemma for cayley 19376. (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 6901 | . . . 4 ⊢ ((𝐹‘𝑥) = (0g‘𝐻) → ((𝐹‘𝑥)‘ 0 ) = ((0g‘𝐻)‘ 0 )) | |
2 | simpr 483 | . . . . . . 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 18929 | . . . . . . . 8 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋) |
6 | 5 | adantr 479 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋) |
7 | cayleylem1.f | . . . . . . . 8 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 7, 3 | grplactval 19005 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = (𝑥 + 0 )) |
9 | 2, 6, 8 | syl2anc 582 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = (𝑥 + 0 )) |
10 | cayleylem1.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
11 | 3, 10, 4 | grprid 18932 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥 + 0 ) = 𝑥) |
12 | 9, 11 | eqtrd 2768 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥)‘ 0 ) = 𝑥) |
13 | 3 | fvexi 6916 | . . . . . . . 8 ⊢ 𝑋 ∈ V |
14 | cayleylem1.h | . . . . . . . . 9 ⊢ 𝐻 = (SymGrp‘𝑋) | |
15 | 14 | symgid 19363 | . . . . . . . 8 ⊢ (𝑋 ∈ V → ( I ↾ 𝑋) = (0g‘𝐻)) |
16 | 13, 15 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ 𝑋) = (0g‘𝐻) |
17 | 16 | fveq1i 6903 | . . . . . 6 ⊢ (( I ↾ 𝑋)‘ 0 ) = ((0g‘𝐻)‘ 0 ) |
18 | fvresi 7188 | . . . . . . 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 2782 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((0g‘𝐻)‘ 0 ) = 0 ) |
21 | 12, 20 | eqeq12d 2744 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((𝐹‘𝑥)‘ 0 ) = ((0g‘𝐻)‘ 0 ) ↔ 𝑥 = 0 )) |
22 | 1, 21 | imbitrid 243 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (0g‘𝐻) → 𝑥 = 0 )) |
23 | 22 | ralrimiva 3143 | . 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 19374 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
26 | eqid 2728 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | 3, 24, 4, 26 | ghmf1 19207 | . . 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 256 | 1 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ↦ cmpt 5235 I cid 5579 ↾ cres 5684 –1-1→wf1 6550 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 Grpcgrp 18897 GrpHom cghm 19174 SymGrpcsymg 19328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-tset 17259 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-efmnd 18828 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-ghm 19175 df-ga 19248 df-symg 19329 |
This theorem is referenced by: cayley 19376 |
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