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Theorem symgfix2 19448
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 3972 . . 3 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}))
2 ianor 983 . . . . 5 (¬ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿) ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
3 fveq1 6905 . . . . . . 7 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
43eqeq1d 2736 . . . . . 6 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
54elrab 3694 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿))
62, 5xchnxbir 333 . . . 4 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
76anbi2i 623 . . 3 ((𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
81, 7bitri 275 . 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 19419 . . . . . 6 (𝑄𝑃 → ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙)
12 eqeq2 2746 . . . . . . . . . . 11 (𝑙 = 𝐿 → ((𝑄𝑘) = 𝑙 ↔ (𝑄𝑘) = 𝐿))
1312rexbidv 3176 . . . . . . . . . 10 (𝑙 = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝑙 ↔ ∃𝑘𝑁 (𝑄𝑘) = 𝐿))
1413rspcva 3619 . . . . . . . . 9 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → ∃𝑘𝑁 (𝑄𝑘) = 𝐿)
15 eqeq2 2746 . . . . . . . . . . . . . . . 16 (𝐿 = (𝑄𝑘) → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1615eqcoms 2742 . . . . . . . . . . . . . . 15 ((𝑄𝑘) = 𝐿 → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1716notbid 318 . . . . . . . . . . . . . 14 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿 ↔ ¬ (𝑄𝐾) = (𝑄𝑘)))
18 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝐾 = 𝑘 → (𝑄𝐾) = (𝑄𝑘))
1918eqcoms 2742 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑄𝐾) = (𝑄𝑘))
2019necon3bi 2964 . . . . . . . . . . . . . 14 (¬ (𝑄𝐾) = (𝑄𝑘) → 𝑘𝐾)
2117, 20biimtrdi 253 . . . . . . . . . . . . 13 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿𝑘𝐾))
2221com12 32 . . . . . . . . . . . 12 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿𝑘𝐾))
2322pm4.71rd 562 . . . . . . . . . . 11 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿 ↔ (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
2423rexbidv 3176 . . . . . . . . . 10 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
25 rexdifsn 4798 . . . . . . . . . 10 (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿))
2624, 25bitr4di 289 . . . . . . . . 9 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2714, 26syl5ibcom 245 . . . . . . . 8 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2827ex 412 . . . . . . 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 857 . . . 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 410 . 2 (𝐿𝑁 → ((𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
348, 33biimtrid 242 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 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  cdif 3959  {csn 4630  cfv 6562  Basecbs 17244  SymGrpcsymg 19400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-tset 17316  df-efmnd 18894  df-symg 19401
This theorem is referenced by: (None)
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