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Theorem symgmatr01 20279
 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 20278 . . . 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 2611 . . . . . 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 2614 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑖 = 𝐾𝑘 = 𝐾))
65adantr 480 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖 = 𝐾𝑘 = 𝐾))
7 eqeq1 2614 . . . . . . . . . 10 (𝑗 = (𝑄𝑘) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
87adantl 481 . . . . . . . . 9 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
98ifbid 4058 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑗 = 𝐿, 1 , 0 ) = if((𝑄𝑘) = 𝐿, 1 , 0 ))
10 oveq12 6558 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖𝑀𝑗) = (𝑘𝑀(𝑄𝑘)))
116, 9, 10ifbieq12d 4063 . . . . . . 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 3694 . . . . . . . . 9 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → 𝑄𝑃)
15 eqid 2610 . . . . . . . . . . 11 (SymGrp‘𝑁) = (SymGrp‘𝑁)
1615, 1symgfv 17630 . . . . . . . . . 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 6113 . . . . . . . . . 10 (1r𝑅) ∈ V
2321, 22eqeltri 2684 . . . . . . . . 9 1 ∈ V
24 symgmatr01.0 . . . . . . . . . 10 0 = (0g𝑅)
25 fvex 6113 . . . . . . . . . 10 (0g𝑅) ∈ V
2624, 25eqeltri 2684 . . . . . . . . 9 0 ∈ V
2723, 26ifex 4106 . . . . . . . 8 if((𝑄𝑘) = 𝐿, 1 , 0 ) ∈ V
28 ovex 6577 . . . . . . . 8 (𝑘𝑀(𝑄𝑘)) ∈ V
2927, 28ifex 4106 . . . . . . 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 6686 . . . . 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 2612 . . . 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 3031 . . 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 246 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  ifcif 4036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  0gc0g 15923  SymGrpcsymg 17620  1rcur 18324 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:  smadiadetlem0  20286
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