MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symgmatr01 Structured version   Visualization version   GIF version

Theorem symgmatr01 21551
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 21550 . . . 4 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 ))
32imp 410 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 )
4 eqidd 2738 . . . . . 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 2741 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑖 = 𝐾𝑘 = 𝐾))
65adantr 484 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖 = 𝐾𝑘 = 𝐾))
7 eqeq1 2741 . . . . . . . . . 10 (𝑗 = (𝑄𝑘) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
87adantl 485 . . . . . . . . 9 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
98ifbid 4462 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑗 = 𝐿, 1 , 0 ) = if((𝑄𝑘) = 𝐿, 1 , 0 ))
10 oveq12 7222 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖𝑀𝑗) = (𝑘𝑀(𝑄𝑘)))
116, 9, 10ifbieq12d 4467 . . . . . . 7 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))))
1211adantl 485 . . . . . 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 488 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → 𝑘𝑁)
14 eldifi 4041 . . . . . . . . 9 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → 𝑄𝑃)
15 eqid 2737 . . . . . . . . . . 11 (SymGrp‘𝑁) = (SymGrp‘𝑁)
1615, 1symgfv 18772 . . . . . . . . . 10 ((𝑄𝑃𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
1716ex 416 . . . . . . . . 9 (𝑄𝑃 → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
1814, 17syl 17 . . . . . . . 8 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
1918adantl 485 . . . . . . 7 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
2019imp 410 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
21 symgmatr01.1 . . . . . . . . . 10 1 = (1r𝑅)
2221fvexi 6731 . . . . . . . . 9 1 ∈ V
23 symgmatr01.0 . . . . . . . . . 10 0 = (0g𝑅)
2423fvexi 6731 . . . . . . . . 9 0 ∈ V
2522, 24ifex 4489 . . . . . . . 8 if((𝑄𝑘) = 𝐿, 1 , 0 ) ∈ V
26 ovex 7246 . . . . . . . 8 (𝑘𝑀(𝑄𝑘)) ∈ V
2725, 26ifex 4489 . . . . . . 7 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) ∈ V
2827a1i 11 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) ∈ V)
294, 12, 13, 20, 28ovmpod 7361 . . . . 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))))
3029eqeq1d 2739 . . . 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 ))
3130rexbidva 3215 . . 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 ))
323, 31mpbird 260 . 2 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 )
3332ex 416 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 209  wa 399   = wceq 1543  wcel 2110  wrex 3062  {crab 3065  Vcvv 3408  cdif 3863  ifcif 4439  cfv 6380  (class class class)co 7213  cmpo 7215  Basecbs 16760  0gc0g 16944  SymGrpcsymg 18759  1rcur 19516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-tset 16821  df-efmnd 18296  df-symg 18760
This theorem is referenced by:  smadiadetlem0  21558
  Copyright terms: Public domain W3C validator