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Mirrors > Home > MPE Home > Th. List > cayleylem1 | Structured version Visualization version GIF version |
Description: Lemma for cayley 18544. (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 |
---|---|
cayleylem1 | ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cayleylem1.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | cayleylem1.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | eqid 2823 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) | |
4 | 1, 2, 3 | gaid2 18435 | . 2 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) ∈ (𝐺 GrpAct 𝑋)) |
5 | cayleylem1.h | . . 3 ⊢ 𝐻 = (SymGrp‘𝑋) | |
6 | cayleylem1.f | . . . 4 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
7 | oveq12 7167 | . . . . . . 7 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑦) = (𝑔 + 𝑎)) | |
8 | ovex 7191 | . . . . . . 7 ⊢ (𝑔 + 𝑎) ∈ V | |
9 | 7, 3, 8 | ovmpoa 7307 | . . . . . 6 ⊢ ((𝑔 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎) = (𝑔 + 𝑎)) |
10 | 9 | mpteq2dva 5163 | . . . . 5 ⊢ (𝑔 ∈ 𝑋 → (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) |
11 | 10 | mpteq2ia 5159 | . . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) |
12 | 6, 11 | eqtr4i 2849 | . . 3 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎))) |
13 | 1, 5, 12 | galactghm 18534 | . 2 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) ∈ (𝐺 GrpAct 𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
14 | 4, 13 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Grpcgrp 18105 GrpHom cghm 18357 GrpAct cga 18421 SymGrpcsymg 18497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-tset 16586 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-subg 18278 df-ghm 18358 df-ga 18422 df-symg 18498 |
This theorem is referenced by: cayleylem2 18543 cayley 18544 |
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