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Theorem symgfix2 17551
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 3454 . . 3 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}))
2 ianor 507 . . . . 5 (¬ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿) ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
3 fveq1 5986 . . . . . . 7 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
43eqeq1d 2516 . . . . . 6 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
54elrab 3235 . . . . 5 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿))
62, 5xchnxbir 321 . . . 4 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
76anbi2i 725 . . 3 ((𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
81, 7bitri 262 . 2 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)))
9 pm2.21 118 . . . . 5 𝑄𝑃 → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
10 symgfix2.p . . . . . . 7 𝑃 = (Base‘(SymGrp‘𝑁))
1110symgmov2 17528 . . . . . 6 (𝑄𝑃 → ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙)
12 eqeq2 2525 . . . . . . . . . . 11 (𝑙 = 𝐿 → ((𝑄𝑘) = 𝑙 ↔ (𝑄𝑘) = 𝐿))
1312rexbidv 2938 . . . . . . . . . 10 (𝑙 = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝑙 ↔ ∃𝑘𝑁 (𝑄𝑘) = 𝐿))
1413rspcva 3184 . . . . . . . . 9 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → ∃𝑘𝑁 (𝑄𝑘) = 𝐿)
15 eqeq2 2525 . . . . . . . . . . . . . . . 16 (𝐿 = (𝑄𝑘) → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1615eqcoms 2522 . . . . . . . . . . . . . . 15 ((𝑄𝑘) = 𝐿 → ((𝑄𝐾) = 𝐿 ↔ (𝑄𝐾) = (𝑄𝑘)))
1716notbid 306 . . . . . . . . . . . . . 14 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿 ↔ ¬ (𝑄𝐾) = (𝑄𝑘)))
18 fveq2 5987 . . . . . . . . . . . . . . . 16 (𝐾 = 𝑘 → (𝑄𝐾) = (𝑄𝑘))
1918eqcoms 2522 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → (𝑄𝐾) = (𝑄𝑘))
2019necon3bi 2712 . . . . . . . . . . . . . 14 (¬ (𝑄𝐾) = (𝑄𝑘) → 𝑘𝐾)
2117, 20syl6bi 241 . . . . . . . . . . . . 13 ((𝑄𝑘) = 𝐿 → (¬ (𝑄𝐾) = 𝐿𝑘𝐾))
2221com12 32 . . . . . . . . . . . 12 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿𝑘𝐾))
2322pm4.71rd 664 . . . . . . . . . . 11 (¬ (𝑄𝐾) = 𝐿 → ((𝑄𝑘) = 𝐿 ↔ (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
2423rexbidv 2938 . . . . . . . . . 10 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿)))
25 rexdifsn 4167 . . . . . . . . . 10 (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿 ↔ ∃𝑘𝑁 (𝑘𝐾 ∧ (𝑄𝑘) = 𝐿))
2624, 25syl6bbr 276 . . . . . . . . 9 (¬ (𝑄𝐾) = 𝐿 → (∃𝑘𝑁 (𝑄𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2714, 26syl5ibcom 233 . . . . . . . 8 ((𝐿𝑁 ∧ ∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙) → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
2827ex 448 . . . . . . 7 (𝐿𝑁 → (∀𝑙𝑁𝑘𝑁 (𝑄𝑘) = 𝑙 → (¬ (𝑄𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
2928com13 85 . . . . . 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 392 . . . 4 ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → (𝑄𝑃 → (𝐿𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3231com13 85 . . 3 (𝐿𝑁 → (𝑄𝑃 → ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿)))
3332impd 445 . 2 (𝐿𝑁 → ((𝑄𝑃 ∧ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
348, 33syl5bi 230 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 194  wo 381  wa 382   = wceq 1474  wcel 1938  wne 2684  wral 2800  wrex 2801  {crab 2804  cdif 3441  {csn 4028  cfv 5689  Basecbs 15579  SymGrpcsymg 17512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-om 6834  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-1o 7323  df-oadd 7327  df-er 7505  df-map 7622  df-en 7718  df-dom 7719  df-sdom 7720  df-fin 7721  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-2 10834  df-3 10835  df-4 10836  df-5 10837  df-6 10838  df-7 10839  df-8 10840  df-9 10841  df-n0 11048  df-z 11119  df-uz 11428  df-fz 12066  df-struct 15581  df-ndx 15582  df-slot 15583  df-base 15584  df-plusg 15665  df-tset 15671  df-symg 17513
This theorem is referenced by: (None)
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