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

Theorem submaval 22587
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 22586 . . 3 (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
543ad2ant1 1134 . 2 ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))
6 simp2 1138 . . 3 ((𝑀𝐵𝐾𝑁𝐿𝑁) → 𝐾𝑁)
7 simpl3 1194 . . 3 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ 𝑘 = 𝐾) → 𝐿𝑁)
81, 3matrcl 22416 . . . . . . . . 9 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
98simpld 494 . . . . . . . 8 (𝑀𝐵𝑁 ∈ Fin)
10 diffi 9215 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑘}) ∈ Fin)
119, 10syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑘}) ∈ Fin)
12 diffi 9215 . . . . . . . 8 (𝑁 ∈ Fin → (𝑁 ∖ {𝑙}) ∈ Fin)
139, 12syl 17 . . . . . . 7 (𝑀𝐵 → (𝑁 ∖ {𝑙}) ∈ Fin)
1411, 13jca 511 . . . . . 6 (𝑀𝐵 → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
15143ad2ant1 1134 . . . . 5 ((𝑀𝐵𝐾𝑁𝐿𝑁) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
1615adantr 480 . . . 4 (((𝑀𝐵𝐾𝑁𝐿𝑁) ∧ (𝑘 = 𝐾𝑙 = 𝐿)) → ((𝑁 ∖ {𝑘}) ∈ Fin ∧ (𝑁 ∖ {𝑙}) ∈ Fin))
17 mpoexga 8102 . . . 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 4636 . . . . . . 7 (𝑘 = 𝐾 → {𝑘} = {𝐾})
2019difeq2d 4126 . . . . . 6 (𝑘 = 𝐾 → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
2120adantr 480 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑘}) = (𝑁 ∖ {𝐾}))
22 sneq 4636 . . . . . . 7 (𝑙 = 𝐿 → {𝑙} = {𝐿})
2322difeq2d 4126 . . . . . 6 (𝑙 = 𝐿 → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
2423adantl 481 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑁 ∖ {𝑙}) = (𝑁 ∖ {𝐿}))
25 eqidd 2738 . . . . 5 ((𝑘 = 𝐾𝑙 = 𝐿) → (𝑖𝑀𝑗) = (𝑖𝑀𝑗))
2621, 24, 25mpoeq123dv 7508 . . . 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, 27ovmpodv2 7591 . 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 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  {csn 4626  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  Basecbs 17247   Mat cmat 22411   subMat csubma 22582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-en 8986  df-fin 8989  df-nn 12267  df-slot 17219  df-ndx 17231  df-base 17248  df-mat 22412  df-subma 22583
This theorem is referenced by:  submaeval  22588  1marepvsma1  22589  smadiadet  22676  submat1n  33804  submatres  33805  madjusmdetlem1  33826
  Copyright terms: Public domain W3C validator