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Theorem symgfix2 19385
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.)
Hypothesis
Ref Expression
symgfix2.p 𝑃 = (Base‘(SymGrp‘𝑁))
Assertion
Ref Expression
symgfix2 (𝐿𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
Distinct variable groups:   𝑘,𝑁   𝑄,𝑘   𝑘,𝐾,𝑞   𝑘,𝐿,𝑞   𝑃,𝑞   𝑄,𝑞
Allowed substitution hints:   𝑃(𝑘)   𝑁(𝑞)

Proof of Theorem symgfix2
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 eldif 3959 . . 3 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}))
2 ianor 979 . . . . 5 (¬ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿) ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
3 fveq1 6901 . . . . . . 7 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
43eqeq1d 2730 . . . . . 6 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
54elrab 3684 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿))
62, 5xchnxbir 332 . . . 4 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
76anbi2i 621 . . 3 ((𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
81, 7bitri 274 . 2 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
9 pm2.21 123 . . . . 5 𝑄𝑃 → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
10 symgfix2.p . . . . . . 7 𝑃 = (Base‘(SymGrp‘𝑁))
1110symgmov2 19356 . . . . . 6 (𝑄𝑃 → ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙)
12 eqeq2 2740 . . . . . . . . . . 11 (𝑙 = 𝐿 → ((𝑄𝑘) = 𝑙 ↔ (𝑄𝑘) = 𝐿))
1312rexbidv 3176 . . . . . . . . . 10 (𝑙 = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝑙 ↔ ∃𝑘𝑁 (𝑄𝑘) = 𝐿))
1413rspcva 3609 . . . . . . . . 9 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → ∃𝑘𝑁 (𝑄𝑘) = 𝐿)
15 eqeq2 2740 . . . . . . . . . . . . . . . 16 (𝐿 = (𝑄𝑘) → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1615eqcoms 2736 . . . . . . . . . . . . . . 15 ((𝑄𝑘) = 𝐿 → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1716notbid 317 . . . . . . . . . . . . . 14 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿 ↔ ¬ (𝑄𝐾) = (𝑄𝑘)))
18 fveq2 6902 . . . . . . . . . . . . . . . 16 (𝐾 = 𝑘 → (𝑄𝐾) = (𝑄𝑘))
1918eqcoms 2736 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑄𝐾) = (𝑄𝑘))
2019necon3bi 2964 . . . . . . . . . . . . . 14 (¬ (𝑄𝐾) = (𝑄𝑘) → 𝑘𝐾)
2117, 20biimtrdi 252 . . . . . . . . . . . . 13 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿𝑘𝐾))
2221com12 32 . . . . . . . . . . . 12 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿𝑘𝐾))
2322pm4.71rd 561 . . . . . . . . . . 11 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿 ↔ (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
2423rexbidv 3176 . . . . . . . . . 10 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
25 rexdifsn 4802 . . . . . . . . . 10 (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿))
2624, 25bitr4di 288 . . . . . . . . 9 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2714, 26syl5ibcom 244 . . . . . . . 8 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2827ex 411 . . . . . . 7 (𝐿𝑁 → (∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙 → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
2928com13 88 . . . . . 6 (¬ (𝑄𝐾) = 𝐿 → (∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3011, 29syl5 34 . . . . 5 (¬ (𝑄𝐾) = 𝐿 → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
319, 30jaoi 855 . . . 4 ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3231com13 88 . . 3 (𝐿𝑁 → (𝑄𝑃 → ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3332impd 409 . 2 (𝐿𝑁 → ((𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
348, 33biimtrid 241 1 (𝐿𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 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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