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Theorem symgmatr01 20221
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 20220 . . . 4 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 ))
32imp 443 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 )
4 eqidd 2610 . . . . . 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 2613 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑖 = 𝐾𝑘 = 𝐾))
65adantr 479 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖 = 𝐾𝑘 = 𝐾))
7 eqeq1 2613 . . . . . . . . . 10 (𝑗 = (𝑄𝑘) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
87adantl 480 . . . . . . . . 9 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
98ifbid 4057 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑗 = 𝐿, 1 , 0 ) = if((𝑄𝑘) = 𝐿, 1 , 0 ))
10 oveq12 6536 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖𝑀𝑗) = (𝑘𝑀(𝑄𝑘)))
116, 9, 10ifbieq12d 4062 . . . . . . 7 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))))
1211adantl 480 . . . . . 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 475 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → 𝑘𝑁)
14 eldifi 3693 . . . . . . . . 9 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → 𝑄𝑃)
15 eqid 2609 . . . . . . . . . . 11 (SymGrp‘𝑁) = (SymGrp‘𝑁)
1615, 1symgfv 17576 . . . . . . . . . 10 ((𝑄𝑃𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
1716ex 448 . . . . . . . . 9 (𝑄𝑃 → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
1814, 17syl 17 . . . . . . . 8 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
1918adantl 480 . . . . . . 7 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → (𝑘𝑁 → (𝑄𝑘) ∈ 𝑁))
2019imp 443 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
21 symgmatr01.1 . . . . . . . . . 10 1 = (1r𝑅)
22 fvex 6098 . . . . . . . . . 10 (1r𝑅) ∈ V
2321, 22eqeltri 2683 . . . . . . . . 9 1 ∈ V
24 symgmatr01.0 . . . . . . . . . 10 0 = (0g𝑅)
25 fvex 6098 . . . . . . . . . 10 (0g𝑅) ∈ V
2624, 25eqeltri 2683 . . . . . . . . 9 0 ∈ V
2723, 26ifex 4105 . . . . . . . 8 if((𝑄𝑘) = 𝐿, 1 , 0 ) ∈ V
28 ovex 6555 . . . . . . . 8 (𝑘𝑀(𝑄𝑘)) ∈ V
2927, 28ifex 4105 . . . . . . 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 6664 . . . . 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 2611 . . . 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 3030 . . 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 245 . 2 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 )
3534ex 448 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 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  {crab 2899  Vcvv 3172  cdif 3536  ifcif 4035  cfv 5790  (class class class)co 6527  cmpt2 6529  Basecbs 15641  0gc0g 15869  SymGrpcsymg 17566  1rcur 18270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-plusg 15727  df-tset 15733  df-symg 17567
This theorem is referenced by:  smadiadetlem0  20228
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