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Mirrors > Home > MPE Home > Th. List > symgfix2 | Structured version Visualization version GIF version |
Description: If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.) |
Ref | Expression |
---|---|
symgfix2.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
Ref | Expression |
---|---|
symgfix2 | ⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3959 | . . 3 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) | |
2 | ianor 979 | . . . . 5 ⊢ (¬ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿) ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿)) | |
3 | fveq1 6901 | . . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾)) | |
4 | 3 | eqeq1d 2730 | . . . . . 6 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿)) |
5 | 4 | elrab 3684 | . . . . 5 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿)) |
6 | 2, 5 | xchnxbir 332 | . . . 4 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿)) |
7 | 6 | anbi2i 621 | . . 3 ⊢ ((𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))) |
8 | 1, 7 | bitri 274 | . 2 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))) |
9 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝑄 ∈ 𝑃 → (𝑄 ∈ 𝑃 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))) | |
10 | symgfix2.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
11 | 10 | symgmov2 19356 | . . . . . 6 ⊢ (𝑄 ∈ 𝑃 → ∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙) |
12 | eqeq2 2740 | . . . . . . . . . . 11 ⊢ (𝑙 = 𝐿 → ((𝑄‘𝑘) = 𝑙 ↔ (𝑄‘𝑘) = 𝐿)) | |
13 | 12 | rexbidv 3176 | . . . . . . . . . 10 ⊢ (𝑙 = 𝐿 → (∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙 ↔ ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿)) |
14 | 13 | rspcva 3609 | . . . . . . . . 9 ⊢ ((𝐿 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙) → ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿) |
15 | eqeq2 2740 | . . . . . . . . . . . . . . . 16 ⊢ (𝐿 = (𝑄‘𝑘) → ((𝑄‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = (𝑄‘𝑘))) | |
16 | 15 | eqcoms 2736 | . . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑘) = 𝐿 → ((𝑄‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = (𝑄‘𝑘))) |
17 | 16 | notbid 317 | . . . . . . . . . . . . . 14 ⊢ ((𝑄‘𝑘) = 𝐿 → (¬ (𝑄‘𝐾) = 𝐿 ↔ ¬ (𝑄‘𝐾) = (𝑄‘𝑘))) |
18 | fveq2 6902 | . . . . . . . . . . . . . . . 16 ⊢ (𝐾 = 𝑘 → (𝑄‘𝐾) = (𝑄‘𝑘)) | |
19 | 18 | eqcoms 2736 | . . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐾 → (𝑄‘𝐾) = (𝑄‘𝑘)) |
20 | 19 | necon3bi 2964 | . . . . . . . . . . . . . 14 ⊢ (¬ (𝑄‘𝐾) = (𝑄‘𝑘) → 𝑘 ≠ 𝐾) |
21 | 17, 20 | biimtrdi 252 | . . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑘) = 𝐿 → (¬ (𝑄‘𝐾) = 𝐿 → 𝑘 ≠ 𝐾)) |
22 | 21 | com12 32 | . . . . . . . . . . . 12 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → ((𝑄‘𝑘) = 𝐿 → 𝑘 ≠ 𝐾)) |
23 | 22 | pm4.71rd 561 | . . . . . . . . . . 11 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → ((𝑄‘𝑘) = 𝐿 ↔ (𝑘 ≠ 𝐾 ∧ (𝑄‘𝑘) = 𝐿))) |
24 | 23 | rexbidv 3176 | . . . . . . . . . 10 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿 ↔ ∃𝑘 ∈ 𝑁 (𝑘 ≠ 𝐾 ∧ (𝑄‘𝑘) = 𝐿))) |
25 | rexdifsn 4802 | . . . . . . . . . 10 ⊢ (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿 ↔ ∃𝑘 ∈ 𝑁 (𝑘 ≠ 𝐾 ∧ (𝑄‘𝑘) = 𝐿)) | |
26 | 24, 25 | bitr4di 288 | . . . . . . . . 9 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)) |
27 | 14, 26 | syl5ibcom 244 | . . . . . . . 8 ⊢ ((𝐿 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙) → (¬ (𝑄‘𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)) |
28 | 27 | ex 411 | . . . . . . 7 ⊢ (𝐿 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙 → (¬ (𝑄‘𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))) |
29 | 28 | com13 88 | . . . . . 6 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))) |
30 | 11, 29 | syl5 34 | . . . . 5 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (𝑄 ∈ 𝑃 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))) |
31 | 9, 30 | jaoi 855 | . . . 4 ⊢ ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → (𝑄 ∈ 𝑃 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))) |
32 | 31 | com13 88 | . . 3 ⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ 𝑃 → ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))) |
33 | 32 | impd 409 | . 2 ⊢ (𝐿 ∈ 𝑁 → ((𝑄 ∈ 𝑃 ∧ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)) |
34 | 8, 33 | biimtrid 241 | 1 ⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3430 ∖ cdif 3946 {csn 4632 ‘cfv 6553 Basecbs 17189 SymGrpcsymg 19335 |
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-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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-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 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-tset 17261 df-efmnd 18835 df-symg 19336 |
This theorem is referenced by: (None) |
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