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Theorem symgmatr01 20508
Description: Applying a permutation that does not fix a certain element of a set to a second element to an index of a matrix a row with 0's and a 1. (Contributed by AV, 3-Jan-2019.)
Hypotheses
Ref Expression
symgmatr01.p 𝑃 = (Base‘(SymGrp‘𝑁))
symgmatr01.0 0 = (0g𝑅)
symgmatr01.1 1 = (1r𝑅)
Assertion
Ref Expression
symgmatr01 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ))
Distinct variable groups:   𝑘,𝑞,𝐿   𝑘,𝐾,𝑞   𝑘,𝑀   𝑘,𝑁   𝑃,𝑘,𝑞   𝑄,𝑘,𝑞   𝑖,𝑗,𝑘,𝑞,𝐿   𝑖,𝐾,𝑗   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗   𝑄,𝑖,𝑗   1 ,𝑖,𝑗,𝑘   0 ,𝑖,𝑗,𝑘
Allowed substitution hints:   𝑅(𝑖,𝑗,𝑘,𝑞)   1 (𝑞)   𝑀(𝑞)   𝑁(𝑞)   0 (𝑞)

Proof of Theorem symgmatr01
StepHypRef Expression
1 symgmatr01.p . . . . 5 𝑃 = (Base‘(SymGrp‘𝑁))
21symgmatr01lem 20507 . . . 4 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 ))
32imp 444 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 )
4 eqidd 2652 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
5 eqeq1 2655 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑖 = 𝐾𝑘 = 𝐾))
65adantr 480 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖 = 𝐾𝑘 = 𝐾))
7 eqeq1 2655 . . . . . . . . . 10 (𝑗 = (𝑄𝑘) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
87adantl 481 . . . . . . . . 9 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
98ifbid 4141 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑗 = 𝐿, 1 , 0 ) = if((𝑄𝑘) = 𝐿, 1 , 0 ))
10 oveq12 6699 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖𝑀𝑗) = (𝑘𝑀(𝑄𝑘)))
116, 9, 10ifbieq12d 4146 . . . . . . 7 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))))
1211adantl 481 . . . . . 6 (((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) ∧ (𝑖 = 𝑘𝑗 = (𝑄𝑘))) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))))
13 simpr 476 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → 𝑘𝑁)
14 eldifi 3765 . . . . . . . . 9 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → 𝑄𝑃)
15 eqid 2651 . . . . . . . . . . 11 (SymGrp‘𝑁) = (SymGrp‘𝑁)
1615, 1symgfv 17853 . . . . . . . . . 10 ((𝑄𝑃𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
1716ex 449 . . . . . . . . 9 (𝑄𝑃 → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
1814, 17syl 17 . . . . . . . 8 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
1918adantl 481 . . . . . . 7 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
2019imp 444 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
21 symgmatr01.1 . . . . . . . . . 10 1 = (1r𝑅)
22 fvex 6239 . . . . . . . . . 10 (1r𝑅) ∈ V
2321, 22eqeltri 2726 . . . . . . . . 9 1 ∈ V
24 symgmatr01.0 . . . . . . . . . 10 0 = (0g𝑅)
25 fvex 6239 . . . . . . . . . 10 (0g𝑅) ∈ V
2624, 25eqeltri 2726 . . . . . . . . 9 0 ∈ V
2723, 26ifex 4189 . . . . . . . 8 if((𝑄𝑘) = 𝐿, 1 , 0 ) ∈ V
28 ovex 6718 . . . . . . . 8 (𝑘𝑀(𝑄𝑘)) ∈ V
2927, 28ifex 4189 . . . . . . 7 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) ∈ V
3029a1i 11 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) ∈ V)
314, 12, 13, 20, 30ovmpt2d 6830 . . . . 5 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))))
3231eqeq1d 2653 . . . 4 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → ((𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ↔ if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 ))
3332rexbidva 3078 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → (∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ↔ ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 ))
343, 33mpbird 247 . 2 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 )
3534ex 449 1 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  ifcif 4119  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  0gc0g 16147  SymGrpcsymg 17843  1rcur 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-tset 16007  df-symg 17844
This theorem is referenced by:  smadiadetlem0  20515
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