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

Theorem submaval 20435
Description: Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
Hypotheses
Ref Expression
submafval.a 𝐴 = (𝑁 Mat 𝑅)
submafval.q 𝑄 = (𝑁 subMat 𝑅)
submafval.b 𝐵 = (Base‘𝐴)
Assertion
Ref Expression
submaval ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝑄(𝑖,𝑗)

Proof of Theorem submaval
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submafval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 submafval.q . . . 4 𝑄 = (𝑁 subMat 𝑅)
3 submafval.b . . . 4 𝐵 = (Base‘𝐴)
41, 2, 3submaval0 20434 . . 3 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
543ad2ant1 1102 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
6 simp2 1082 . . 3 ((𝑀𝐵𝐾𝑁𝐿𝑁) → 𝐾𝑁)
7 simpl3 1086 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
81, 3matrcl 20266 . . . . . . . . 9 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 474 . . . . . . . 8 (𝑀𝐵𝑁 ∈ Fin)
10 diffi 8233 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑘}) ∈ Fin)
119, 10syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑘}) ∈ Fin)
12 diffi 8233 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑙}) ∈ Fin)
139, 12syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑙}) ∈ Fin)
1411, 13jca 553 . . . . . 6 (𝑀𝐵 → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
15143ad2ant1 1102 . . . . 5 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
1615adantr 480 . . . 4 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
17 mpt2exga 7291 . . . 4 (((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) ∈ V)
1816, 17syl 17 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) ∈ V)
19 sneq 4220 . . . . . . 7 (𝑘 = 𝐾 → {𝑘} = {𝐾})
2019difeq2d 3761 . . . . . 6 (𝑘 = 𝐾 → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
2120adantr 480 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
22 sneq 4220 . . . . . . 7 (𝑙 = 𝐿 → {𝑙} = {𝐿})
2322difeq2d 3761 . . . . . 6 (𝑙 = 𝐿 → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
2423adantl 481 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
25 eqidd 2652 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑀𝑗) = (𝑖𝑀𝑗))
2621, 24, 25mpt2eq123dv 6759 . . . 4 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
2726adantl 481 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
286, 7, 18, 27ovmpt2dv2 6836 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗))))
295, 28mpd 15 1 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  {csn 4210  cfv 5926  (class class class)co 6690  cmpt2 6692  Fincfn 7997  Basecbs 15904   Mat cmat 20261   subMat csubma 20430
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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
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-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-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-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-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-fin 8001  df-slot 15908  df-base 15910  df-mat 20262  df-subma 20431
This theorem is referenced by:  submaeval  20436  1marepvsma1  20437  smadiadet  20524  submat1n  29999  submatres  30000  madjusmdetlem1  30021
  Copyright terms: Public domain W3C validator