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Theorem symgfix2 17659
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 3550 . . 3 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}))
2 ianor 508 . . . . 5 (¬ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿) ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
3 fveq1 6102 . . . . . . 7 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
43eqeq1d 2612 . . . . . 6 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
54elrab 3331 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿))
62, 5xchnxbir 322 . . . 4 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
76anbi2i 726 . . 3 ((𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
81, 7bitri 263 . 2 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
9 pm2.21 119 . . . . 5 𝑄𝑃 → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
10 symgfix2.p . . . . . . 7 𝑃 = (Base‘(SymGrp‘𝑁))
1110symgmov2 17636 . . . . . 6 (𝑄𝑃 → ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙)
12 eqeq2 2621 . . . . . . . . . . 11 (𝑙 = 𝐿 → ((𝑄𝑘) = 𝑙 ↔ (𝑄𝑘) = 𝐿))
1312rexbidv 3034 . . . . . . . . . 10 (𝑙 = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝑙 ↔ ∃𝑘𝑁 (𝑄𝑘) = 𝐿))
1413rspcva 3280 . . . . . . . . 9 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → ∃𝑘𝑁 (𝑄𝑘) = 𝐿)
15 eqeq2 2621 . . . . . . . . . . . . . . . 16 (𝐿 = (𝑄𝑘) → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1615eqcoms 2618 . . . . . . . . . . . . . . 15 ((𝑄𝑘) = 𝐿 → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1716notbid 307 . . . . . . . . . . . . . 14 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿 ↔ ¬ (𝑄𝐾) = (𝑄𝑘)))
18 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝐾 = 𝑘 → (𝑄𝐾) = (𝑄𝑘))
1918eqcoms 2618 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑄𝐾) = (𝑄𝑘))
2019necon3bi 2808 . . . . . . . . . . . . . 14 (¬ (𝑄𝐾) = (𝑄𝑘) → 𝑘𝐾)
2117, 20syl6bi 242 . . . . . . . . . . . . 13 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿𝑘𝐾))
2221com12 32 . . . . . . . . . . . 12 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿𝑘𝐾))
2322pm4.71rd 665 . . . . . . . . . . 11 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿 ↔ (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
2423rexbidv 3034 . . . . . . . . . 10 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
25 rexdifsn 4264 . . . . . . . . . 10 (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿))
2624, 25syl6bbr 277 . . . . . . . . 9 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2714, 26syl5ibcom 234 . . . . . . . 8 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2827ex 449 . . . . . . 7 (𝐿𝑁 → (∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙 → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
2928com13 86 . . . . . 6 (¬ (𝑄𝐾) = 𝐿 → (∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3011, 29syl5 33 . . . . 5 (¬ (𝑄𝐾) = 𝐿 → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
319, 30jaoi 393 . . . 4 ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3231com13 86 . . 3 (𝐿𝑁 → (𝑄𝑃 → ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3332impd 446 . 2 (𝐿𝑁 → ((𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
348, 33syl5bi 231 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 195  wo 382  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  {crab 2900  cdif 3537  {csn 4125  cfv 5804  Basecbs 15695  SymGrpcsymg 17620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-tset 15787  df-symg 17621
This theorem is referenced by: (None)
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