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Theorem symgmatr01 22676
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 22675 . . . 4 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 ))
32imp 406 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 1 , 0 ), (𝑘𝑀(𝑄𝑘))) = 0 )
4 eqidd 2736 . . . . . 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 2739 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑖 = 𝐾𝑘 = 𝐾))
65adantr 480 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖 = 𝐾𝑘 = 𝐾))
7 eqeq1 2739 . . . . . . . . . 10 (𝑗 = (𝑄𝑘) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
87adantl 481 . . . . . . . . 9 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑗 = 𝐿 ↔ (𝑄𝑘) = 𝐿))
98ifbid 4554 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → if(𝑗 = 𝐿, 1 , 0 ) = if((𝑄𝑘) = 𝐿, 1 , 0 ))
10 oveq12 7440 . . . . . . . 8 ((𝑖 = 𝑘𝑗 = (𝑄𝑘)) → (𝑖𝑀𝑗) = (𝑘𝑀(𝑄𝑘)))
116, 9, 10ifbieq12d 4559 . . . . . . 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 484 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → 𝑘𝑁)
14 eldifi 4141 . . . . . . . . 9 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → 𝑄𝑃)
15 eqid 2735 . . . . . . . . . . 11 (SymGrp‘𝑁) = (SymGrp‘𝑁)
1615, 1symgfv 19412 . . . . . . . . . 10 ((𝑄𝑃𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
1716ex 412 . . . . . . . . 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 406 . . . . . 6 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘𝑁) → (𝑄𝑘) ∈ 𝑁)
21 symgmatr01.1 . . . . . . . . . 10 1 = (1r𝑅)
2221fvexi 6921 . . . . . . . . 9 1 ∈ V
23 symgmatr01.0 . . . . . . . . . 10 0 = (0g𝑅)
2423fvexi 6921 . . . . . . . . 9 0 ∈ V
2522, 24ifex 4581 . . . . . . . 8 if((𝑄𝑘) = 𝐿, 1 , 0 ) ∈ V
26 ovex 7464 . . . . . . . 8 (𝑘𝑀(𝑄𝑘)) ∈ V
2725, 26ifex 4581 . . . . . . 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 7585 . . . . 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 2737 . . . 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 3175 . . 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 257 . 2 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 )
3332ex 412 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 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  ifcif 4531  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  0gc0g 17486  SymGrpcsymg 19401  1rcur 20199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-tset 17317  df-efmnd 18895  df-symg 19402
This theorem is referenced by:  smadiadetlem0  22683
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