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Theorem symgfix2 18522
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 3920 . . 3 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}))
2 ianor 979 . . . . 5 (¬ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿) ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
3 fveq1 6642 . . . . . . 7 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
43eqeq1d 2823 . . . . . 6 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
54elrab 3657 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿))
62, 5xchnxbir 336 . . . 4 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
76anbi2i 625 . . 3 ((𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
81, 7bitri 278 . 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 18494 . . . . . 6 (𝑄𝑃 → ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙)
12 eqeq2 2833 . . . . . . . . . . 11 (𝑙 = 𝐿 → ((𝑄𝑘) = 𝑙 ↔ (𝑄𝑘) = 𝐿))
1312rexbidv 3283 . . . . . . . . . 10 (𝑙 = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝑙 ↔ ∃𝑘𝑁 (𝑄𝑘) = 𝐿))
1413rspcva 3598 . . . . . . . . 9 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → ∃𝑘𝑁 (𝑄𝑘) = 𝐿)
15 eqeq2 2833 . . . . . . . . . . . . . . . 16 (𝐿 = (𝑄𝑘) → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1615eqcoms 2829 . . . . . . . . . . . . . . 15 ((𝑄𝑘) = 𝐿 → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1716notbid 321 . . . . . . . . . . . . . 14 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿 ↔ ¬ (𝑄𝐾) = (𝑄𝑘)))
18 fveq2 6643 . . . . . . . . . . . . . . . 16 (𝐾 = 𝑘 → (𝑄𝐾) = (𝑄𝑘))
1918eqcoms 2829 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑄𝐾) = (𝑄𝑘))
2019necon3bi 3033 . . . . . . . . . . . . . 14 (¬ (𝑄𝐾) = (𝑄𝑘) → 𝑘𝐾)
2117, 20syl6bi 256 . . . . . . . . . . . . 13 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿𝑘𝐾))
2221com12 32 . . . . . . . . . . . 12 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿𝑘𝐾))
2322pm4.71rd 566 . . . . . . . . . . 11 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿 ↔ (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
2423rexbidv 3283 . . . . . . . . . 10 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
25 rexdifsn 4700 . . . . . . . . . 10 (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿))
2624, 25syl6bbr 292 . . . . . . . . 9 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2714, 26syl5ibcom 248 . . . . . . . 8 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2827ex 416 . . . . . . 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 854 . . . 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 414 . 2 (𝐿𝑁 → ((𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
348, 33syl5bi 245 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 209  wa 399  wo 844   = wceq 1538  wcel 2115  wne 3007  wral 3126  wrex 3127  {crab 3130  cdif 3907  {csn 4540  cfv 6328  Basecbs 16461  SymGrpcsymg 18473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-tset 16562  df-efmnd 18012  df-symg 18474
This theorem is referenced by: (None)
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