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Mirrors > Home > MPE Home > Th. List > symgextf | Structured version Visualization version GIF version |
Description: The extension of a permutation, fixing the additional element, is a function. (Contributed by AV, 6-Jan-2019.) |
Ref | Expression |
---|---|
symgext.s | ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) |
symgext.e | ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) |
Ref | Expression |
---|---|
symgextf | ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁⟶𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 774 | . . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ 𝑥 = 𝐾) → 𝐾 ∈ 𝑁) | |
2 | simpllr 775 | . . . . 5 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ ¬ 𝑥 = 𝐾) → 𝑍 ∈ 𝑆) | |
3 | simpr 484 | . . . . . . 7 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁) | |
4 | neqne 2945 | . . . . . . 7 ⊢ (¬ 𝑥 = 𝐾 → 𝑥 ≠ 𝐾) | |
5 | 3, 4 | anim12i 612 | . . . . . 6 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ ¬ 𝑥 = 𝐾) → (𝑥 ∈ 𝑁 ∧ 𝑥 ≠ 𝐾)) |
6 | eldifsn 4791 | . . . . . 6 ⊢ (𝑥 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑥 ∈ 𝑁 ∧ 𝑥 ≠ 𝐾)) | |
7 | 5, 6 | sylibr 233 | . . . . 5 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ ¬ 𝑥 = 𝐾) → 𝑥 ∈ (𝑁 ∖ {𝐾})) |
8 | eqid 2728 | . . . . . 6 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾})) | |
9 | symgext.s | . . . . . 6 ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
10 | 8, 9 | symgfv 19333 | . . . . 5 ⊢ ((𝑍 ∈ 𝑆 ∧ 𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑍‘𝑥) ∈ (𝑁 ∖ {𝐾})) |
11 | 2, 7, 10 | syl2anc 583 | . . . 4 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ ¬ 𝑥 = 𝐾) → (𝑍‘𝑥) ∈ (𝑁 ∖ {𝐾})) |
12 | 11 | eldifad 3959 | . . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) ∧ ¬ 𝑥 = 𝐾) → (𝑍‘𝑥) ∈ 𝑁) |
13 | 1, 12 | ifclda 4564 | . 2 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁) → if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)) ∈ 𝑁) |
14 | symgext.e | . 2 ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) | |
15 | 13, 14 | fmptd 7124 | 1 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁⟶𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 ifcif 4529 {csn 4629 ↦ cmpt 5231 ⟶wf 6544 ‘cfv 6548 Basecbs 17179 SymGrpcsymg 19320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-tset 17251 df-efmnd 18820 df-symg 19321 |
This theorem is referenced by: symgextf1 19375 symgextfo 19376 symgextres 19379 |
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